The thirteen books of Euclid's Elements

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THE THIRTEEN BOOKS

OF

EUCLID'S ELEMENTS

TRANSLATED FROM THE TEXT OF HEIBERG

WITH INTRODUCTION AND COMMENTARY

BY
T. L. HEATH, C.B., Sc.D.,

SOMBTIMK FELLOW OF TRINmr COLLKGB, CAMBMDGB

VOLUME I
INTRODUCTION AND BOOKS 1, H

* • . •

Cambridge :
at the University Press

1908

1*1

I

THE THIRTEEN BOOKS

•OF

EUCLID'S ELEMENTS

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE

C. F. CLAY, Manager.

Eiltei: FETTER LANE, E.C.

•Mrtml: too. PRINCES STREET.

Ikrlte: A. ASHSR AND 00.
; P. A. BROCKHAUS,
G. P. PUTNAM'S SON&
n» Calcvlte: IfACMILLAN AND CO., Lm

[Al/ X^Jk/s merved.]

A***r

XT rT»«

•K(^i«4P4« 'tjV*^ -HmP V^ fttC J? . ^ l^if. iVTi fi- uj

Chap.

I.

. »»

II.

»>

III.

>»

IV.

»»

V.

»«

VI.

M

VII.

t *>

VIII.

5

f "

IX.

CONTENTS

VOLUME I.

INTRODUCTION.

p.

eucud and the traditions about him

Euclid's other works

Greek commentators other than Proclus

Proclus and his sources

The Text

The Scholia

Euclid in Arabia

PRiNaPAL translations and editions .

§ I. On the nature op Elements i

§ 2. Elements anterior to Euclid's . . . :
I §3. First principles: Depinitiqns, Postulates

AND Axioms . . . ' . i

[ § 4. Theorems and Problems . . . . j

( § 5. The formal divisions of a proposition i

h § 6. Other technical terms . . . . :

§ 7. The definitions i

[ THE ELEMENTS.

I Book I. Definitions, Postulates, Common Notions . ]

Notes on Definitions etc. ]

Propositions . j

Book II. Definitions

Note on geometrical algebra
Propositions . ^ . . .

—

•—

k

X

CONTENTS
VOLUME II.

Book

IIL

Definitions .

Propositions ,

Book

IV,

Definitions
Propositions ,

Book

V.

Introductory hote
Definitjons ,

Propositions ,

Book

VL

Introductory hote

Definitions .
Propositions .

Book

VIL

Definitions

Propositions ,

Book

VIIL

• 1

Book ;IX,

*

Greek Index to Vol, H. .

FjfGLiSH Ihdex. to Vol. 1L

FAOt

t

?8

119

Its

138

18S
191

«77
99$

345
384

431

VOLUME IlL

Book X. Introductory note i

Definitions 10

Propositions i^ — ^47 . , . , • ,14-101

Definitions il ,/,.... ioi

Propositions 4S — 84 iq?-i77

Definitions hi ryj

Propositions 85—115 17&-254

Ancient extensions of theory of Book X . 255

Book XL Definitions . * a6o

Propositions . - - ' 27a

Book XIL Historical note . . < . ' • . . 365

Propositions 369

Book XIII* Historical note 43S

Propositions ......*. 44a

iPPENDix. L The so-called "Book XI V," (by Hypsicles) 511

IL Note on the so-called **Book XV/* , , 519

iddknda et corrigenda .*...... £21

* Index: Greek 329

English 535

]

n

ifir Christoph Schlussel rtad Christoph KUn

ERRATA

Vol. I. p. i9» line \*i^fcr ** bat not a platfonn and sixpence " rtid " bat not a figare and
tizpence"

» P> io5» luM lO

»• p* 415* col* <• Une 17
Vol. II. p. 41, line 97
Vol. III. p. 539, col. I, line 45
VoL I. p. 106, line i \

99 P* 415* col. I, line ai y/or Giacomo read Giovanni
VoL III. p. 558, coL a, line 95 j
VoL I. p. 18a, line %\^ for Opot read^Opn
Vol. III. p. 499. In the 6gnre» CV ihoald be a thick line.

iTM'l EUCUD

PREFACE

\"nnHERE never has been, and till we see it we nev<
^ X shall believe that there can be, a system of geometi
worthy of the name, which has any material departures (we d
not speak of corrections or extensions or developments) froi
the plan laid down by Euclid," De Morgan wrote thus i
October 1848 (Short supplementary remarks on the first s\
Books of EuclicTs Elements in the Companion to the Almam
for 1849); and I do not think that, if he had been livin
to-day, he would have seen reason to revise the opinion s
deliberately pronounced sixty years ago. It is true that in th
interval much valuable work has been done on the continei
in the investigation of the first principles, including th

. formulation and classification of axioms or postulates whic
are necessary to make good the deficiencies of Euclid's ow
explicit postulates and axioms and to justify the furth<
assumptions which he tacitly makes in certain proposition
content apparently to let their truth be inferred from observ;
tion of the figures as drawn ; but, once the first principles ai
disposed of, the body of doctrine contained in the recent tex
books of elementary geometry does not, and from the natui
of the case cannot, show any substantial differences from th;
set forth in the Elements. In England it would seem that f<
less of scientific value has been done ; the efforts of a multitud
of writers have rather been directed towards producing alte;
natives for Euclid which shall be more suitable, that is to sa;
easier, for schoolboys. It is of course not surprising that, i

vi PREFACE

these days of short cuts, there should have arisen a movement
to get rid of Euclid and to substitute a *' royal road to
geometry " ; the marvel is that a book v^^hich was not written
for schoolboys but for grown men (as all internal evidence
shows^ and in particular the essentially theoretical character
of the work and its aloofness from anything of the nature of
" practical " geometry) should have held its own as a school-
book for so long» And now that Euclid's proofs and arrange-
ment are no longer required from candidates at examinations
there has been a rush of competitors anxious to be first in the
field with a new text-book on the more " practical " lines which
now find so much favour. The natural desire of each teacher
who writes such a text- book is to give prominence to some
special nostrum which he has found successful with pupils.
One result is, too often, a loss of a due sense of proportion ;
and, in any case, it is inevitable that there should be great
diversity of treatment. It was with reference to such a danger
that Lardner wrote in 1846 : '* Euclid once superseded, every
teacher would esteem his own work the best, and every school
would have its own class book. All that rigour and exactitude
which have so long excited the admiration of men of science
would be at an end. These very words would lose all definite
meaning. Every school would have a different standard ;
matter of assumption in one being matter of demonstration in
another; until, at length, Geometry, in the ancient sense of
the word, would be altogether frittered away or be only
considergd as a particular application of Arithmetic and
Algebra/j It is, perhaps, too early yet to prophesy what will
be the ultimate outcome of the new order of things ; but it
would at least seem possible that history will repeat itself and
that, when chaos has come again in geometrical teaching,
there will be a return to Euclid more or less complete for the
purpose of standardising it once more.

But the case for a new edition of Euclid is independent of
any controversies as to how geometry shall be taught to
schoolboys. Euclid's work will live long after all the text-books

PREFACE

(of the present day are superseded and forgotten. It is on
of the noblest monuments of antiquity ; no mathematicia
worthy of the name can afford not to know Euclid, the r«
Euclid as distinct from any revised or rewritten version
which will serve for schoolboys or engineers. And, to kno^
L Euclid, it is necessary to know his language, and, so far as
' can be traced, the history of the "elements" which h
; collected in his immortal work.

\ This brings me to the raison d'Hre of the present editioi

A new translation from the Greek was necessary for tw
I reasons. First, though some time has elapsed since th
'^ appearance of Heiberg's definitive text and prolegomens
I published between 1883 and 1888, there has not been, so fa
^ as I know, any attempt to make a faithful translation from
I into English even of the Books which are commonly reac
^ And, secondly, the other Books, vii. to x. and xiii., were nc
' included by Simson and the editors who followed him, c
: apparently in any English translation since Williamson
(178 1 — 8), so that they are now practically inaccessible t
English readers in any form.

In the matter of notes, the edition of the first six Book

' in Greek and Latin with notes by Camerer and Haubc

(Berlin, 1824 — 5) is a perfect mine of information. It woul

h have been practically impossible to make the notes mor

\ exhaustive at the time when they were written. But th

researches of the last thirty or forty years into the history <

mathematics (I need only mention such names as those <

r Bretschneider, Hankel, Moritz Cantor, Hultsch, Paul Tannery

j^ Zeuthen, Loria, and Heiberg) have put the whole subjec

i( upon a different plane. I have endeavoured in this editio

to take account of all the main results of these researches u

to the present date. Thus, so far as the geometrical Book

^ are concerned, my notes are intended to form a sort c

(f dictionary of the history of elementary geometry, arrange

according to subjects ; while the notes on the arithmetic*

Books VII. — IX. and on Book x. follow the same plan.

viii PREFACE

I desire to express here my thanks to my brother,
Dr R. S* Heath, Vice- Principal of Birmingham University,
for suggestions on the proof sheets and, in particular p for the
reference to the parallelism between Euclid's definition of
proportion and Dedekind's theory of irrationals, to Mr R. D*
Hicks for advice on a number of difficult points of translation,
to Professor A. A. Bevan for help in the transliteration of
Arabic names, and to the Curators and Librarian of the
Bodleian Library for permission to reproduce, as frontispiece,
a page from the famous Bodleian MS. of the Ehments^
Lastly, my best acknowledgments are due to the Syndics of
the Cambridge University Press for their ready acceptance
of the work, and for the zealous and efficient cooperation of
their staff which has much lightened the labour of seeing the
book through the Press,

T. U H,
NovembcTy 1908.

X

• • •

r . ■.•••:

• • •

^ INTRODUCTION.

I

CHAPTER I.

EUCLID AND THE TRADITIONS ABOUT HIM.

As in the case of the other great mathematicians of Greece, so in
Euclid's case, we have only the most meagre particulars of the life
and personality of the man^

Most of what we have is contained in the passage of Proclus'
summary relating to him, which is as follows^:

" Not much younger than these (sc. Hermotimus of Colophon and
Philippus of Mende) is Euclid, who put together the Elements, collect-
ing many of Eudoxus' theorems, perfecting many of Theaetetus', and
also bringing to irrefragable demonstration the things which were
only somewhat loosely proved by his predecessors. This man lived*
in Uie time of the first Ptolemy. For Archimedes, who came imme-
diately after the first (Ptolemy)', makes mention of Euclid: and,
further, they say that Ptolemy once asked him if there was in
geometry any shorter way than that of the elements, and he answered
that there was no royal road to geometry*. He is then younger than
the pupils of Plato but older than Eratosthenes and Archimedes ; for
the latter were contemporary with one another, as Eratosthenes some-
where says."

This passage shows that even Proclus had no direct knowledge
^ of Euclici s birthplace or of the date of his birth or death. He pro-
- ceeds by inference. Since Archimedes lived just after the first

I ^ Proclus, ed. Friedlein, p. 68, 6 — lo.

' * The word yfywe must apparently mean ''flourished," as Heiberg understands it

^ ^LitterargeschichtNche StudUn iiber Euklid, 1881, p. 16), not "was bom," as Hankel took

it : otherwise part of Proclus' argument would lose its cogency.
^ 'So Heiberg understands /iri/SoXiiw r^ wptimfi (sc. nroXefuUy). Friedlein's text has

I Kol between iwifioKunf and nf wptirtfi; and it is rignt to remark that another reading is
U KolipTif wpilbrtf (without irtfiaXtitf) which has been translated " in his first ^>(," by wmch
is understood On the Sphere and Cylmder i., where (i) in Prop- ^ ure the words *' let BC
be made equal to D by the seamd (proposition) of the first of Euclid's (books)," and (a) in
Prop. 6 the words " For these things are handed down in the Elements " (without the name
of Euclid). Heiberg thinks the former passage is rdPerred to, and tnat Proclus must
therefore have had before him the words " by the second of the first of Euclid ** : a fair proof
that they are genuine, though in themselves &iey would be somewhat suspicious,
f ^ The same story is told in Stobaeus, Eel, (11. p. 338, 30, ed. Wachsmuth) about

Alexander and Menaechmus. Alexander is represented as having asked Menaechmiis to
teach him geometry concisely, but he replied : '*0 kinff, through the country there are royal
roads and roads for common citixens, but in geometry there is one road for all.*'

H. E. "^ I

t

f

y

4

t u/'", INTRODUCTION [gh j

Ptolemy/'ftoS Archimedes mentions Euclid^ while there is an anecdote
dhout'J^f^e Ptolemy and Euclid, therefore EucUd hVed in the time of
the fii^l Ptolemy.

.W^'may infer then from Proclus that Euclid was intermediate

between the first pupils of Plato and Archimedes. Now Plato died in

■3^ J Archimedes lived 287-212, Eratosthenes 276-194 B.C Thus

r, Ej/clid must have flourished c. 300 B,C, which date agrees well with

V*the fact that Ptolemy reigned from 306 to 2 S3 B.C

/•■ It is most probable that Euclid received his mathematical training

* in Athens from the pupils of Plato; for most of the geometers who
could have taught him were of that school, and it was in Athens that
the older writers of elements, and the other mathematicians on whose
works Euclid's Elemtnts depend, had lived and taught He may
himself have been a Platonist, but this does not follow from the state-
ments of Proclus on the subject. Proclus says namely that he was of
the school of Plato and in close touch with that philosophy \ But
this was only an attempt of a New Platonist to connect Euclid with
his philosophy I as is clear from the next words in the same sentence,
•* for which reason also he set before himself, as the end of the whole
Elements, the construction of the so-called Platonic figures" It is
evident that it was only an idea of Proclus* own to infer that Euclid
was a Platonist because his Elemmts end with the investigation of
the five regular solids, since a later passage shows him hard put to
it to reconcile the view that the construction of the five regular solids
was the end and aim of the Eiements with the obvious fact that they
were intended to supply a foundation for the study of geometry in
general, "to make perfect the understanding of the learner in regard -
to the whole of geometry'" To get out of the difficulty he says' that,
if one should ask him what was the aim (<rico7ro9) of the treatise, he
would reply by making a distinction between Euclid's intentions
(i) as regards the subjects with which his investigations are concerned,
(2) as regards the learner, and would say as regards (I) that "the
whole of the geometer's argument is concerned with the cosmic
figures." This latter statement is obviously incorrect It is true
that Euclid's Elements end with the construction of the five regular
solids; but the plan i metrical portion has no direct relation to them,
and the arithmetical no relation at all ; the propositions about them
are merely the conclusion of the stereo metrical division of the work.

One thing is however certain, namely that Euclid taught, and
founded a school, at Alexandria, This is clear from the remark of
Pappus about Apollonius*: *'he spent a very long time with the
pupils of Euclid at Alexandria, and it was thus that he acquired
such a scientific habit of thought**

It is in the same passage that Pappus makes a remark which
might J to an unwary reader, seem to throw some light on the

^ Proclus, p. ^1 90, Kol rf rpoatp^ct £j nXorbuvtr&f iart jral rj ^X«<ra^9 Tm:CfrQ alxiUitt.

■ (W^, p* 71, 8. * ibid. p» 70, 19 ^qq,

* PttppU^, VII. p* ISjSt 10— iIt inAT^oXd^at r«r itwh £l^j(X«iJcM^ pud^itmlt ir 'AXtfardpeffi

I

CH. i] EUCLID AND TRADITIONS ABOUT HIM

personality of Euclid. He is speaking about Apollonius' prefac

to the first book of his Conies^ where he says that Euclid had no

I completely worked out the synthesis of the "three- and four-Iin<

r* locus," which in fact was not possible without some theorems firs

discovered by himself. Pappus says on this^: "Now Euclid-

"^ regarding Aristaeus as deserving credit for the discoveries he ha(

already made in conies, and without anticipating him or wishing t<

construct ane\v the same system (such was his scrupulous fairness an(

V his exemplary kindliness towards all who could advance mathematica

' science to however small an extent), being moreover in no wise con

tentious and, though exact, yet no braggart like the other [ApoUonius

. — ^wrote so much about the locus as was possible by means of th<

• conies of Aristaeus, without claiming completeness for his demonstra

tions." It is however evident, when the passage is examined in it

context, that Pappus is not following any tradition in giving thi

^ account of Euclid: he was offended by the terms of Apolloniuj

' reference to Euclid, which seemed to him unjust, and he drew \

\ fancy picture of Euclid in order to show ApoUonius in a relativel]

^ unfavourable light
Another story is told of Euclid which one would like to believe true

I According to^tobaeus*, " some one who had begun to read geometr
with EuclidT^en he had learnt the first theorem, asked Euclid, ' Bu

f what shall 1 get by learning these things ? ' Euclid called his slav

\ and said ' Give him threepence, since he must make gain out of wha

j he learns.'"

i In the middle ages most translators and editors spoke of Eucli(

as Euclid of Megara, This description arose out of a confusioi
between our Euclid and the philosopher Euclid of Megara who livec
about 400 B.C The first trace of this confusion appears in Valeriu
Maximus (in the time of Tiberius) who says' that Plato, on beinj
appealed to for a solution of the problem of doubling the cubica
altar, sent the inquirers to "Euclid the geometer." There is no doub
about the reading, although an early commentator on Valeriu
Maximus wanted to correct "Eucliden" into **EudoxufnP and thi
correction is clearly right. But, if Valerius Maximus took Euclid th(
geometer for a contemporary of Plato, it could only be througl
confusing him with Euclid of Megara. The first specific reference t
Euclid as Euclid of Megara belongs to the 14th century, occurring i:
the virofunjfiarur/jLoi of Theodorus Metochita (d. 1332) who speaks c
" Euclid of Megara, the Socratic philosopher, contemporary of Plato,
as the author of treatises on plane and solid geometry, data, optic
etc. : and a Paris MS. of the 14th century has " Euclidis philosoph
Socratici liber elementorum.'' The misunderstanding was genera
in the period from Campanus' translation (Venice 1482) to those c
Tartaglia (Venice 1565) and Candalla (Paris 1566). But on
Constantinus Lascaris (d. about 1493) had already made the prope

^ Pappus, VII. pp. 676, 45 — 678, 6. Hultsch, it is true, brackets the whole possag
pp. 676, 95—^78, 15, but apparently on the ground of the diction only.
* Stobaeus, /,e. * viii. la, ext. i.

^m

4 INTRODUCTION [cH. i

distinction by saying of our Euclid that " he was different from him
of Megara of whom Laertius wrote, and who wrote dialogues*'' ; and
to Commandinus belongs the credit of being the first translator' to
put the matter beyond doubt : *' Let us then free a number of people
from the error by which they have been induced to believe that our
Euclid is the same as the philosopher of Megara" etc

Another idea, that Euclid was born at Gela in Sicily, is due to the
same confusion, being based on Diogenes Laertius' description' of the
philosopher Euclid as being *'of Megara, or, according to some, of
Gela, as Alexander says in the ^ea^o^^atV

In view of the poverty of Greek tradition on the subject even as
early as the time of Proclus (410-485 a.d,), we must necessarily take
a*m grano the apparently circumstantial accounts of Euclid given by
Arabian authors ; and indeed the origin of their stories can b^
explained as the result (i) of the Arabian tendency to romancep and
(2) of misunderstandings.

We read* that " Euclid, son of Naucrates, grandson of Zenarchus*
called the author of geometry, a philosopher of somewhat ancient
date, a Greek by nationality domiciled at Damascus, bom at Tyre,
most learned in the science of geometry, published a most excellent
and most useful work entitled the foundation or elements of geometry,
a subject in which no more general treatise existed before among the
Greeks :fnay, there was no one even of later date who did not walk
in his footsteps and frankly profess his doctrine. Hence ako Greek,
Roman and Arabian geometers not a few, who undertook the task
of illustrating this work, published commentaries, scholia, and notes
upon it, and made an abridgment of the work itself For this reason
the Greek philosophers used to post up on the doors of their schools
the weli*known notice : * Let no one come to our school, who has not
first learned the elements of Euclid.*"! The details at the beginning
of this extract cannot be derived from Greek sources, for even Proclus
did not know anything about Euclid's father, while it was not the
Greek habit to record the names of grandfathers, as the Arabians
commonly did. Damascus and Tyre were no doubt brought in to
gratify a desire which the Arabians always showed to connect famous
Greeks in some way or other with the East Thus Naslraddlnj the
translator of the Elemenis^ who was of Tus in Khurasan, actually
makes Euclid out to have been "Thusinus" also* The readiness of
the Arabians to run away with an idea is illustrated by the last words

^ Letter to Fe maud us Acuna^ printed in Mauroljcus, HUiima SuiHa^f foL 91 r> (see
Heiberg, Evklid-Studien^ pp. ^i — 3, 45),
' Prcf»ce to translation (risauri, 'S?^)*

* Diog. L. II. io6t p* 58 ed. Cobet.

< Casiri, BiMiptkeca Arabka^Hispema SicuriaUmiiy I. p. 339* CasiH's source i» *1-
Qlftl (d- 1343), the author of the Tt^rikh al-HukamSy a collection of biographies of phi-
losophers, mathematicians, a^tronotncrs eta

* The Fihrist says *'son of Naucratei, the son of Berenice (?) " {set Suter's translatLOO in
Ahkandlungm ittr Gts<:h. d. Math. VI. Heft, 1893, p, 16 J,

' The same predilection made the Ambs de^ritie Pythagoras as a P^P^^ ^^ ^^^ ^^
Salome, Hipmrchus as the exponent of Chaldaean philosophy or as ibe Chaldaean, Archi-
medes as an Egyptian etc. (H&ji Khalfaj Lexicon Sibiic^apkuum, and Casiri).

CH. i] EUCLID AND TRADITIONS ABOUT HIM 5

of the extract Everyone knows the story of Plato's inscription over
the porch of the Academy : " let no one unversed in geometry enter
my doors " ; the Arab turned geometry into Euclid s geometry, and
told the story of Greek philosophers in general and 'Hheir Academies."

Equally remarkable are the Arabian accounts of the relation of
Euclid and Apollonius\ According to them the Elements were
originally written, not by Euclid, but by a man whose name was
Apollonius, a carpenter, who wrote the work in 15 books or sections*.
In the course of time some of the work was lost and the rest became
disarranged, so that one of the kings at Alexandria who desired to
study geometry and to master this treatise in particular first questioned
about it certain learned men who visited him and then sent for Euclid
who was at that time famous as a geometer, and asked him to revise
and complete the work and reduce it to order. Euclid then re-wrote
it in 13 books which were thereafter known by his name. (According
to another version Euclid composed the 13 books out of commentaries
which he had published on two books of Apollonius on conies and
out of introductory matter added to the doctrine of the five regular
solids.) To the thirteen books were added two more books, the work
of others (though some attribute these also to Euclid) which contain
several things not mentioned by Apollonius. According to another
version Hypsicles, a pupil of Euclid at Alexandria, offered to the
king and published Books xiv. and xv., it being also stated that
Hypsicles had "discovered" the books, by which it appears to be
suggested that Hypsicles had edited them from materials left by Euclid.

We observe here the correct statement that Books XI v. and XV.
were not written by Euclid, but along with it the incorrect informa-
tion that Hypsicles, the author of Book xiv., wrote Book xv. also.

The whole of the fable about Apollonius having preceded Euclid
and having written the Elements appears to have been evolved out of
the preface to Book xiv. by Hypsicles, and in this way ; the Book
must in early times have been attributed to Euclid, and the inference
based upon this assumption was left uncorrected afterwards when it
was recognised that Hypsicles was the author. The preface is worth
quoting :

" Basilides of Tyre, O Protarchus, when he came to Alexandria
and met my father, spent the greater part of his sojourn with him on
account of their common interest in mathematics. And once, when

^ The authorities for these statements (quoted by Casiri and Hajl Khalfa are al-KindI*s
tract di imtituto Uhri Euclidis (al-Kindl died about 87^) and a commentary by (^ftd1z3de
ar-RumI (d. about '440) on a book called Ashkdl at-ta: sis (fundamental propositions) by
Ashraf Shamsaddin as-Samarqandl (^. 1176) consisting of elucidations of 35 propositions
selected from the first books of Euclid. Na^Iraddln likewise says that Euclid cut out two of
15 books of elements then existing and published the rest under his own name. According to
Q&dlz&de the king heard that there was a celebrated geometer named Euclid at Tyre: Naflr-
adciln says that he sent for Euclid of Tus.

* So says the Fihrist, Suter (op! cit, p. 49) thinks that the author of the Fihrist did not
suppose Apollonius of Perga to be the writer of the EUtnents^ as later Arabian authorities
did, but that he distinguished another Apollonius whom he odls **a carpenter." Sutor's
argument is based on the fact that the Fihrisfs article on Apollonius (of Perga) says nothins;
of the Elements^ and that it gives the three great mathematicians, Euclid, Archimedes and
Apollonius, in the correct chronological order.

K

A INTRODUCTION [cH. i

examining the treatise written by Apollonius about the comparison
between the dodecahedron and the icosahedron inscribed in the same
sphere, (showing) what ratio they have to one another, they thought
that Apollonius had not expounded this matter properly, and
accordingly they emended the exposition, as I was able to learn
from my father. And I myself, later, fell in with another book
published by Apollonius, containing a demonstration relating to the
subject, and I was greatly interested in the investigation of the
problem. The book published by Apollonius is accessible to all—
for it has a large circulation, having apparently been carefully written
out later — ^but I decided to send you the comments which seem to
me to be necessary, for you will through your proficiency in mathe-
matics in general and in geometry in particular form an expert
judgment on what I am about to say, and you will lend a kindly ear
to my disquisition for the sake of your friendship to my father and
your goodwill to me."

The idea that Apollonius preceded Euclid must evidently have
been derived from the passage just quoted. It explains other things
besides, Basilides must have been confused with ffatrtX^vs, and we
have a probable explanation of the "Alexandrian king,'* and of the
"learned men who visited'* Alexandria. It is possible also that in
the '* Tyrian '' of Hypsicles' preface we have the origin of the notion
that Euclid was bom in Tyre, These inferences argue, no doubt,
very defective knowledge of Greek : but we could expect no better
from those who took the Organon of Aristotle to be " instrumentum
musicum pneumaticum/' and who explained the name of Euclid,
which they variously pronounced as Uclid^s or Icludes, to be com-
pounded of Ucli a key, and Dis a measure, or, as some say, geometry,
so that Udides is equivalent to the k^ of geametfyl

Lastly the alternative version, given in brackets above, which says
that Euclid made the Eiimenis out of commentaries which he wrote
on two books of Apollonius on conies and prolegomena added to the
doctrine of the five solids, seems to have arisen, through a like
confusion, out of a later passage' in Hypsicles* Book xiv, : "And this
is expounded by Aristaeus in the book entitled 'Comparison of the five
figures,' and by Apollonius in the second edition of his comparison of
the dodecahedron with the icosahedron.'* The "doctrine of the five
solids " in the Arabic must be the " Comparison of the five figures "
in the passage of Hypsicles, for nowhere else have we any information
about a work bearing this title, nor can the Arabians have had. The
reference to the tivo books of Apollonius on conks will then be the
result of mixing up the fact that Apollonius wrote a book on conies
with the second edition of the other work mentioned by Hypsicles.
We do not find elsewhere in Arabian authors any mention of a
commentary by Euclid on Apollonius and Aristaeus : so that the
story in the passage quoted is really no more than a variation of the
fable that the Elements were the work of Apollonius,

^ Heiberg's Eaclld, toL v* p. 6,

f

CHAPTER II.

y EUCLID'S OTHER WORKS.

In giving a list of the Euclidean treatises other than the EUmeHts,
I shall be brief: for fuller accounts of them, or speculations with
regard to them, reference should be made to the standard histories of
mathematics ^

I will take first the works which are mentioned by Greek authors.

I. The Pseudaria,

I mention this first because Proclus refers to it in the general
remarks in praise of the Elements which he gives immediately after
the mention of Euclid in his summary. He says': "But, inasmuch
as many things, while appearing to rest on truth and to follow from
scientific principles, really tend to lead one astray from the principles
and deceive the more superficial minds, he has handed down methods
for the discriminative understanding of these things as well, by the
use of which methods we shall be able to give beginners in this study
practice in the discovery of paralogisms, and to avoid being misled.
This treatise, by which he puts tiiis machinery in Our hands, he
entitled (the book) of Pseudaria, enumerating in order their various
kinds, exercising our intelligence in each case by theorems of all
sorts, setting the true side by side with the false, and combining
the refutation of error with practical illustration. This book then is
by way of cathartic and exercise, while the Elements contain the
irrefragable and complete guide to the actual scientific investigation
of the subjects of geometry."

The book is considered to be irreparably lost We may conclude
however from the connexion of it with the Elenunts and the reference
to its usefulness for beginners that it did not go outside the domain
of elementary geometry*.

^ Heibeig gives very exhaustive details in his LitterargeschichtlUhe Studien iiber Euklid\
the best of the shorter accounts are those of Cantor (Gesch, d. Afaih. i,, 1907, pp. 378 — 394)
and Loria (II periodo aureo delta geometria greca^ p. 9 and pp. 63 — 85).

• Proclus, p. 70, I— 18.

' Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his
commentary on Aristotle's Sophistici EUnchi (fol. 15 bV. **Not only those (AeTxoc) which do
not start from the principles of the science, under which the problem is classed... but also
those which do start from the proper principles of the science but in some respect admit a
paralogism, e.g. the Fseudographanata of Euclid." Tannery (BuU. des sciences math, et astr,
1* S^e, vi., 1883, i^ Partie, p. 147) conjectures that it may be from this treatise that the
same commentator got his information about the quadratures of the circle by Antiphon and

INTRODUCTION [cH. ii

r

2. The Data

The Data {Z€topi,iifa) are included by Pappus in the Treasury af
Analysis {tqitok ava\v6^^o^\ and he describes their contents*. They
are still concerned with elementary geometry, though forming part
of the introduction to higher analysis. Their form is that of pro-
positions proving that, if certain things in a figure are given (in
magnitude, in species, etc.)* something else is given. The subject*
matter Is much the same as that of the plani metrical books of the \
Elements, to which the Data are often supplementary. We shall see
this later when we come to compare the propositions in the Elements
which give us the means of solving the general quadratic equation
with the corresponding propositions of the Data which give the
solution. The Data may in fact be regarded as elementary exercises
in analysis.

It is not necessary to go more closely into the contents, as we
have the full Greek text and the commentary by Marinus newly
edited by Menge and therefore easily accessible*

3- The book On divisions {ofjignres).

This work {ir^pl Biaif>icr€wv fit^iov) is mentioned by Proclus*.
In one place he is speaking of the conception or definition {X0709)
o{ figure, and of the divisibility of a figure into others differing from
it in kind ; and he adds: "For the circle is divisible into parts unlike
in definition or notion {dvofiGta t^ >Jrf^\ and so is each of the
rectilineal figures ; this is in fact the business of the writer of the
Elements in his Divisions, where he divides given figures, in one case
into like figures, and in another into unlike**" "Like" and "unlike"
here mean, not "similar*' and "dissimilar** in the technical sense, but
"like" or "unlike in definition or notion'' (Xo^^): thus to divide a
triangle into triangles would be to divide it into "like" figures, to
divide a triangle into a triangle and a quadrilateral would be to
divide it into "unlike" figures.

The treatise is lost in Greek but has been discovered in the
Arabic. First John Dee discovered a treatise De divisionibus by one
Muhammad Bagdad] n us ^ and handed over a copy of it (in Latin) in
1563 to Commandinus, who published it, in Dee's name and his own,
in IS70*, It was formerly supposed that Dee had himself translated

BrysoD, to say aothing of the lunules of Hippocrates* I tliink however that there b an
objection to this theory so far as regank Brysooi for Alexander distinctly lays that Biyson^s
quadrature did not start from the proper priijcipJes of geometfy, hut frofn some priiuriples
more generat.

" Pappus, vn. p. 638.

' VoL VI ♦ in the Teiibner edition of EuctMii cftra otnnm by Heiberg and Menge. A
translation of the Puta is also included In Simson a EucHd (though culturally his text left
much to be desired)*

■ Proclusj p. 69 J 4. * ihid. J 44, 21— 16-

* Steinschneider places him in the totlic. H. Sv^itt {BthH^ktca Maihtmaika, iVj, 1903,
pp. 34, 17) ideoti^es him with Abu(Bekr) Moh. b. "Abdalbiql al-Ba^adl, QSdl 0ud^ey of
M&rbtiln {drca lojo-n+t), to whom he also attributes the Libtrj^dti (? judicis) super dtamu^
Euelidii translated by Gherard of Cremona.

* Df superfHtrum ditnsionihus liher Math&mdo Bagitadina aditriptui^ nurv primttm
Uannis Dtt Londinensii it Fed^^i C^mfrmttdini Urhinntis eptra itt iu€fm tiMhiSi Plsaurt,
1570, afterwania bduded in Gregory's Euclid {Oxford, 1703)-

CH. ii] EUCLID'S OTHER WORKS 9

the tract into Latin from the Arabic*; but it now appears certain'
that he found it in Latin in a Cotton MS. now in the British Museum.
Dee, in his preface addressed to Commandinus, says nothing of his
having translated the book, but only remarks that the very illegible
MS. had caused him much trouble and (in a later passage) speaks of
••the actual, very ancient, copy from which I wrote out..** (in ipso
unde descripsi vetustissimo exemplari). The Latin translation of
this tract from the Arabic was probably made by Gherard of Cremona
(1114-1187), among the list of whose numerous translations a "liber
divisionum" occurs. The Arabic original cannot have been a direct
translation from Euclid, and probably was not even a direct adapta-
tion of it ; it contains mistakes and unmathematical expressions, and
moreover does not contain the propositions about the division of a
circle alluded to by Proclus. Hence it can scarcely have contained
more than a fragment of Euclid's work.
\ But Woepcke found in a MS. af Paris a treatise in Arabic on the

division of figures, which he translated and published in 1851*. It is
expressly attributed to Euclid in the MS. and corresponds to the
description of it by Proclus. Generally speaking, the divisions are
divisions into figures of the same kind as the original figures, e.g. of
triangles into triangles; but there are also divisions into " unlike*'
figures, e.g. that of a triangle by a straight line parallel to the base.
The missing propositions about the division of a circle are also here :
**to divide into two equal parts a given figure bounded by an arc
of a circle and two straight lines including a given angle" and "to
draw in a given circle two parallel straight lines cutting off a certain
part of the circle." Unfortunately the proofs are given of only four
propositions (including the two last mentioned) out of 36, because
the Arabic translator found them too easy and omitted them. To
illustrate the character of the problems dealt with I need only take
one more example : " To cut off a certain fraction from a (parallel-)
trapezium by a straight line which passes through a given point lying
inside pr outside the trapezium but so that a straight line can be
drawn through it cutting both the parallel sides of the trapezium."
The genuineness of the treatise edited by Woepcke is attested by the
facts that the four proofs which remain are elegant and depend on
propositions in the Elements, and that there is a lemma with a true
Greek ring: "to apply to a straight line a rectangle equal to the
rectangle contained hy AB, AC and deficient by a square'' Moreover
the treatise is no fragment, but finishes with the words " end of the
treatise," and is a well-ordered and compact whole. Hence we may
safely conclude that Woepcke's is .not only Euclid's own work but
the whole of it*. A restoration of the work, with proofs, was attempted

^ Heiberg, Euklid'Studien, p. 13.

' H. Suter in Bibliotkeca MaihematUa, iv,, 1905-6, pp. 331—3.

• Journal Asiatique, 185 1, p. 133 sgq.

* We are told by Casiri that Th&bit b. Qarra emended the translation of the Uber de
divisumibus\ but Ofterdinger seems to be wrong in sajring that according to Gartz (/?/ i$Uer-
prUUms et expianaioribus EuciuUs AraHcis schediasma histancum, HaUe, 1813) there is a

10 INTRODUCTION [ch, h

by OfterdingerS who however does not give Woepckc's props. 30, 31,

34* 35i 3^^

4. The Porisms.

It is not possible to give in this place any account of the con-
troversies about the contents and signiEcance of the three lost books
of Porisms, or of the important attempts by Robert Simson and
Chasles to restore the work. These may be said to form a whole
literature, references to which will be found most abundantly given
by Heiberg and Loria, the former of whom has treated the subject
from the philological point of view; most exhaustively, while the
latter, founding himself generally on Heiberg, has added useful
detailsj from the mathematical side, relating to the attempted restora-
tions, etc,' It must suffice here to give an extract from the only
original source of information about the nature and contents of the
P^nsms, namely Pappus', In his general preface about the books
composing the Treasury of Afmfysis (totto? aWXvd^evo^) he says :

"After the Tangencies (of Apollonius) come, in three books, the
Porisms of Euclid, [in the view of many] a collection most ingeniously
devised for the analysis of the more weighty problems, [and] although
nature presents an unlimited number of such porisms', [they have
added nothing to what was written originally by Euclid, except that
some before my time have shown their want of taste by adding to a
few (of the propositions) second proofs^ each (proposition) admitting
of a definite number of demonstrations, as we have shown, and
Euclid having given one for each, namely that which is the most
lucid* These porlsms embody a theory subtle, natural, necessary,
and of considerable generality, which is fascinating to those >vho can
see and produce results].

** Now all the varieties of porisms belong, neither to theorems nor
problems, but to a species occupying a sort of intermediate position
[so that their enunciations can be formed like those of either theorems
or problems], the result being that, of the great number of geometers,
some regarded them as of the class of theorems, and others of pro-
blems, looking only to the form of the proposition. But that the
ancients knew better the difference between these three things, is
clear from the 'definitions* For they said that a theorem is that
which is proposed with a view to the demonstration of the very
thing proposed J a problem that which is thrown out with a view to
the construction of the very thing proposed, and a porism that which
is proposed with a view to the producing of the very thing proposed.
[But this definition of the porism was changed by the more recent
writers who could not produce everything, but used these elements

CDnapkt« MS. of Tfaabit'9 tmnslAtioQ m ihe Escurml. I cannot find any such staCemcDt in
Gaitz.

^ L. F. Oftcvdingeri Stiirc^ tur WUdfrkersi£liung tUr Sckrifl dei Euklides iiber dk
Tkdilung d^ Figurm^ Ulm, 185^-

' Heiberg, Eukiid-S(udien^ pp, 56^ — 79, uwl Loha, H pftiodo tmrea ddla geometric gt^tif

pp. 70— Si, 13J— 5.

' pAppO£, cd. HuJudi, VII, pp, 648—660. I put in square brackeu the words bracket^
bj HuJtsck

* I adopt Heibefg'a reading of a comma here instead of a full stop* — ^ *

CH. n] EUCLID'S OTHER WORKS ii

and proved only the fact that that which is sought really exists, but
did not produce it^ and were accordingly confuted by the definition
and the whole doctrine. They based their definition on an incidental
characteristic, thus : A porism is that which falls short of a locus-
theorem in respect of its hypothesis*. Of this kind of porisms loci
are a species, and they abound in the Treasury of Analysis ; but
this species has been collected, named and handed down separately
from the porisms, because it is more widely diffused than the other
species]. But it has further become characteristic of porisms that,
owing to their complication, the enunciations are put in a contracted
form, much being by usage left to be understood; so that many
geometers understand them only in a partial way and are ignorant of
the more essential features of their contents.

^[Now to comprehend a number of propositions in one enunciation
is by no means easy in these porisms, because Euclid himself has not
in fact given many of each species, but chosen, for examples, one or a
few out of a great multitude^ But at the beginning of the first book
he has given some propositions, to the number of ten, of one species,
namely that more fruitful species consisting of loci.] Consequently,
finding that these admitted of being comprehended in one enunciation,
we have set it out thus:

If, in a system of four straight lines^ which cut each other
two and two, three points on one straight line be given while the
rest except one lie on different straight lines given in position,
the remaining point also will lie on a straight line given in
position'.

^ Heiberg points out that Props. 5—^ of Archimedes' treatise On Sfirals are porisms in
this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K,
It is then pK)6sible, says Archimedes, to araw a straight line q g

KHF^ meeting the circumference in H and the tangent in /*,

such that

FH\HK<(%szBiri\c,

where c is the circomference of any circle. To prove this he
assumes the following construction. E being any straight line
greater than r, he says : let KG be parallel to DF^ "and let
the line GH equal to i? be placed verging to the point B.^^
Archimedes must of course nave known how to effect this
construction, which requires conies. But that it is possible requires very little argument, for
if we draw any straight line BHG meeting the circle in ZTand KG in G^ it is obvious that
as G moves away from C, HG becomes neater and greater and may be made as great as we
please. The '* later writers *' would no doubt have contented themselves with this considera-
tion without actually constructing HG,

' As Heibere says, this translation is made certain by a preceding passage of Pappus
0>> 648, I — 3) where he compares two enunciations, the latter of which *' falls short of the
former in hypothesis but goes oeyond it in requirement,^'* E.g. the first enunciation requiring
us, given three circles, to draw a circle touching all three, the second may require us, given
only two circles (one less datum), to draw a circle touching them and of a given siie (an
extra requirement).

* I translate Heiberg's reading with a full stop here followed by rpbt dffxv ^ ^h*"^ l""/^
^}dt^ {ifiofjJpop) Hultsch] Tov vpibrou fiipxiov...,

* The four straight lines are described in the text as (the sides) hrrlov 4 wapvwrlov, Le.
sides of two sorts of quadrilaterals which Simson tries to explain (see p. no of the Index
GraecitcUis of Hultsch*s edition of Pappus).

* In other words (Cbasles, p. «3 ; Loria, p. 73) if a triangle be so deformed that each of
its sides turns about one of three points in a straight line, and two of its vertices lie on two
straight lines given in position, the third vertex will klso lie on a straight line.

tJ INTRODUCTION [ch. ii

**Thts has only been enunciated of four straight line*!, of which not
more than two pass through the same point, but it is not known (to
most people) that it is true of any assigned number of straight lines
if enunciated thus:

If any number of straight lines cut one another, not more
than two (passing) through the same point, and all the points
(of intersection situated) on one of them be given, and if each of
those which are on another (of them) lie on a straight line given
in position —
or still more generally thus :

if any number of straight lines cut one another, not more than

two (passing) through the same point, and all the points (of

intersection situated) on one of them be given, while of the other

points of intersection in multitude equal to a triangular number

a number corresponding to the side of this triangular number tie

respectively on straight lines given in position, provided that of

these latter points no three are at the angular points of a triangle

{sc. having for sides three of the given straight lines)— each of the

remaining points will lie on a straight h'ne given in position*.

" It is probable that the writer of the Elements was not unaware

of this but that he only set out the principle ; and he seems, in the

case of all the porisms, to have laid down the principles and the

seed only [of many important things], the kinds of which should be

distinguished according to the differences, not of their hypotheses, but

of the results and the things sought [All the hypotheses are different

from one another because they are entirely special, but each of the

results and things sought, being one and the same, follow from many

different hypotheses.]

"We must then in the first book distinguish the following kinds of
things sought :

"At the beginning of the book* is this proposition :

I, ' If from two givm points straight lines be drawn meeting
on a straight line given in position^ and one cut off from a straight
Urn given in position (a segment measured) to a given point on i/,
the other wilt also cut off from afwther {straight line a segment)
having to the first a given ratio'
*■ Following on this (we have to prove)

IL that such and such a point lies on a straight line given

in position ;
II L that the ratio of such and such a pair of straight lines
is given ; '*
etc. etc. (up to xxix.).
^The three books of the porisms contain 38 lemmas; of the
theorems themselves there are 171."

^ Loria (p. 7^ note) gives the meaning of this as follows, pointing out that Simson was
the discoverer o( it : " Iia complete M-lateral be deformed so that its sides respectively turn
about n points on a straight line, and (it- i) of its n (n- i)/a vertices move on as many
straight Imes, the other (m- i)(ii-9)/a of its vertices likewise move on as many straight
lines: but it is necessary that it shoula be impossible to form with the (m- i) vertices any
triangle having for sides the sides of the polygon."

' Reading, with Keiberg, roH fitfiXlov [roC r Hultsch].

\

CH. ii] EUCUiyS OTHER WORKS 13

Pappus further gives lemmas to the Porisms (pp. 866 — 918, ed,
Hultsch).

With Pappus' account of Porisms must be compared the passages
of Proclus on the same subject Proclus distinguishes two senses in
which the word ir6pur/jui is used. The first is that of corollary where
something appears as an incidental result of a proposition, obtained
without trouble or special seeking, a sort of bonus which the investi-
gation has presented us with^ The other sense is that of Euclid's
Porisf9ts\ In this sense' ^^porism is the name given to things which
are sought, but need some finding and are neither pure bringing into
existence nor simple theoretic argument For (to prove) that the
angles at the base of isosceles triangles are equal is a matter of
theoretic argument, and it is with reference to things existing that
such knowl^ge is (obtained). But to bisect an angle, to construct a
triangle, to cut off, or to place — all these things demand the making
of something ; and to find the centre of a given circle, or to find the
greatest common measure of two given commensurable magnitudes,
or the like, is in some sort between theorems and problems. For in
these cases there is no bringing into existence of the things sought,
but finding of them, nor is the procedure purely theoretic. For it is
necessary to bring that which is sought into view and exhibit it to
the eye. Such are the porisms which Euclid wrote, and arranged in
three books of Porisms.

Proclus' definition thus agrees well enough with the first, ** older,"
definition of Pappus. A porism occupies a place between a theorem
and a problem: it deals with something dXreaidy existing, ^s a theorem
does, but has to find it (e.g. the centre of a circle), and, as a certain
operation is therefore necessary, it partakes to that extent of the
nature of a problem, which requires us to construct or produce some-
thing not previously existing. Thus, besides IIL i of the Elements
and X. 3, 4 mentioned by Proclus, the following propositions are
real porisms: ill. 25, VL 11— 13, vii. 33, 34, 36, 39, vill. 2,4, x. 10,
XIIL 18. Similarly in Archimedes On the Sphere and Cylifuler I. 2 — 6
might be called porisms.

The enunciation given by Pappus as comprehending ten of Euclid's
propositions may not reproduce the form of Euclid's enunciations ;
but, comparing the result to be proved, that certain points lie on
straight lines given in position, with the class indicated by ll. above,
where the question is of such and such a point lying on a straight line
given in position, and with other classes, e.g. (v.) that such and such a
line is given in position, (vi.) that such and such a line verges to a given
point, (XXVII.) that there exists a given point such that straight lines
drawn from it to such and such (circles) will contain a triangle given
in species, we may conclude that a usual form of a porism was ^* to
prove that it is possible to find a point with such and such a property"

* Proclus, pp. a I a, 14 ; 301, «i.

' ibid. p. 313, I a. "The term porism is used of certain problems, like the Porisms
written by Euclid."

* ihid, pp. 301, 35 sqq.

14 INTRODUCTION [CB. ii

or "a straight line on which lie all the points satisfying given
conditions" etc.

Simson defined a porism thus : " Porisma est propositio iti qua
proponitur dem oust rare rem aliquam, vel pi u res datas esse^ cuij vel
quibus, ut et cuilibet ex rebus innumeriSj non quidem datis, sed quae
ad ea quae data sunt eandem habent relationem, con venire ostendendum
est affecttonem quandam communem in propositione descriptamV

From the above it is easy to understand Pappus' statement that
loci constitute a large class of porisms. A hcus is well defined by
Simson thus : '* Locus est propositio in qua propositum est datam
esse demonstrare, vel in venire lineam aut superficiem cuius quodlibet
punctum, vel superficiem in qua quaelibet linea data lege descripta,
communem quandam habet proprietatem in propositione descriptam."
Heiberg cites an excellent instance of a l<fats which is Sip&rism, namely
the following proposition quoted by Eutocius* from the P/afie Zati of
ApoHonius :

*' Given two points in a plane, and a ratio between unequal straight
lines, it is possible to draw, in the plane, a circle such that the straight
lines drawn from the given points to meet on the circumference of
the circle have (to one another) a ratio the same as the given ratio/*

A difficult point, however, arises on the passage of Pappus, which
says that a porism is "that which, in respect of its hypothesis, falls
short of a locus-theorem" (roTrueov ^€6>/j?J^aTo?)- Heiberg explains it
by comparing the porism from ApoHonius' Platit Xa^i just given with
Pappus' enunciation of the same thing, to the effect that, if from two
given points two straight lines be drawn meeting in a point, and these
straight lines have to one another a given ratio, the point will lie on
either a straight line or a circumference of a circle given in position.
Heiberg observes that in this latter enunciation something is taken
into the hypothesis which was not in the hypothesis of the enunciation
of the porism, viz. "that the ratio of the straight lines is the same."
I confess this does not seem to me satisfactory : for there is no real
difference between the enunciations, and the supposed difference in
hypothesis is very like playing with words. Chasles says : " Ct qui
const itu€ U porisme est ct qui manque d I'hypoth^e d'un tASreme
local (en d 'aut res termes, le porisme est infifrieur, par Vhypoth^se, au
th^orimc local; c'est-i*dire que quand quelques parties d'une pro-
position locale n'ont pas dans T^nonc^ la determination qui leur est
propre, cette proposition cesse d'etre regardee com me un th^rfeme et
devient un porisme)/* But the subject still seems to require further
elucidation.

While there is so much that is obscure, it seems certain (i) that the
Porisms were distinctly pait of higher geometry and not of elementary

^ This was thus expressed bj Cbasles i '* Le porisme ^ udc propositioti dons laquellc on
dcmimde de d^montrer qu'unc chose ou plusieurs choses sont d^nn/es, qui, flinsi que iWe
quelcouque d'une in^nUe d^autres choses non donates, mais dont chacune est avec des choses
doDufes daoa u&e m^me relation, one une cert^ine propri^t^ comoiune, d^rite dam Ja pio^
position, "

■ Commentary on ApoUonius* C^ni^s (vol. ii. p, iSo^ ed. Heiberg).

1

I*

CH. II] EUCLID'S OTHER WORKS 15

geometry, (2) that they contained propositions belonging to the
modem theory of transversals and to projective geometry. It should
i be remembered too that it was in the course of his researches on this
subject that Chasles was led to the idea of anharmanic ratios.

Lastly, allusion should be made to the theory of Zeuthen^ on the
subject of the porisms. He observes that the only porism of which
Pappus gives the complete enunciation, *' If from two given points
straight lines be drawn meeting on a straight line given in position,
and one cut off from a straight line given in position (a segment
measured) towards a given point on it, the other will also cut off from
another (straight line a segment) bearing to the first a given ratio,"
is also true if there be substituted for the first given straight line a
conic regarded as the '' locus with respect to four lines," and that this
extended porism can be used for completing Apollonius' exposition
of that locus. Zeuthen concludes that the Porisms were in part by-
products of the theory of conies and in part auxiliary means for the
study of conies, and that Euclid called them by the same name as
that applied to corollaries because they were corollaries with respect to
conies. But there appears to be no evidence to confirm this conjecture.

5. The Surface-loci (tmto* irpo^ em<f>aif€ia).

The two books on this subject are mentioned by Pappus as part
of the Treasury of Analysis^. As the other works in the list which
were on plane subjects dealt only with straight lines, circles, and
conic sections, it is a priori likely that among the loci in this treatise
(loci which are surfaces) were included such loci as were cones,
cylinders and spheres. Beyond this all is conjecture based on two
lemmas given by Pappus in connexion with the treatise.

(i) The first of these lemmas' and the figure attached to it are
not satisfactory as they stand, but a possible restoration is indicated
by Tannery*. If the latter is right, it suggests that one of the loci
contained all the points on the elliptical parallel sections of a cylinder
and was therefore an oblique circular cylinder. Other assumptions
with r^ard to the conditions to which the lines in the. figure may be
subject would suggest that other loci dealt with were cones regarded
as containing all points on particular elliptical parallel sections of
the cones*.

(2) In the second lemma Pappus states and gives a complete proof
of the focus-and-directrix property of a conic, viz. that the locus of a
point whose distance from a given point is in a given ratio to its distance
from a fixed line is a conic section^ which is an ellipse ^ a parabola or a
hyperbola according as the given ratio is less than, equal to, or greater
than unity\ Two conjectures are possible as to the application of
this theorem in Euclid's Surface-loci, (a) It may have been used to
prove that the locus of a point whose distance from a given straight

^ Die Lehre von den Ktgdschnitten im AUertum^ chapter viii.

• Pappus, VII. p. 636. • ibid. vii. p. 1004.

* Bulletin des sciences math, ei astron.., 3* S^rie, vi. 149.

• Further particulars will be found in The IVarks e/ Archimedes, pp. Ixii — Ixiv, and in
Zeuthen, Die Lehre von den JCegeischnitten^ p. 415 sqq.

* Pappus, VII. pp. 1006 — 1014, and Hultsch*s Applendix, pp. 1370 — 3.

i6 INTRODUCTION - [CH. ii

line is in a given ratio to its distance from a given plane is a certain
cone. {6} It may have been used to prove that the locus of a point
whose distance from a given point is in a given ratio to its distance
from a given plane is the surface formed by the revolution of a conic
about its major or conjugate axis^ Thus Chasles may have been
correct in his conjecture that the Surfaa-ioci dealt with surfaces of
revolution of the second degree and sections of the same^

6, The Conks.

Pappus says of this lost work; **The four books of Euclid's Conies
were completed by ApoUonius, who added four more and gave us
eight books of Conies'.*' It is probable that Euclid's work was lost
even by Pappus' time, for he goes on to speak of '* Aristaeus, who wrote
the still extant five books of Solid Loci connected with the conies/'
Speaking of the relation of Euclid's work to that of Aristaeus on conies
regarded as loci ^ Pappus says in a later passage (bracketed however
by Hultsch) that Euclid, regarding Aristaeus as deserving credit for
the discoveries he had already made in conies, did not (try to)
anticipate him or construct anew the same system. We may no
doubt conclude that the book by Aristaeus on solid loci preceded
Euclid's on conies and was, at least in point of originality, more
important Though both treatises dealt with the same subject-matter^
the object and the point of view were different ; had they been the
same, Euclid could scarcely have refrained, as Pappus says he did,
from attempting to improve upon the earlier treatise* No doubt
Euclid wrote on the general theory of conies as Apollonius did, but
confined himself to those properties which were necessary for the
analysis of the Solid Loci of Aristaeus. The Conies of Euclid were
evidently superseded by the treatise of Apollonius,

As regards the contents of Euclid's Conies^ the most important
source of our information is Archimedes, who frequently refers to
propositions in conies as well known and not needing proof, adding
in three cases that they are proved in the " elements of conies '* or in
'*the conies," which expressions must clearly refer to the works of
Aristaeus and Euclid**

Euclid still used the old names for the conies (sections of a right-
angled» acute*angled, or obtuse -angled cone), but he was aware that
an ellipse could be obtained by cutting a cone in any manner by a
plane parallel to the base (assuming the section to lie wholly between
the apex of the cone and its base) and also by cutting a cylinder.
This is expressly stated in a passage from the Phaenomma of Euclid
about to te mentioned*,

7* The Phaenonuna.

This is an astronomical work and is still extant A much inter-

* For further d^taib sec Tlis Worki of Attkimidis, pp, bdv, latv, aod Zeuthen, /♦ f»

* A^fH Misi^riyuff pp, 373—4' * Pappus, vri* p. 671.

* For details of these proposition* see my Apoiknims 0/ Ptrga^ pp. stxiv, xnjtvi.

' See Heitjerg, Mukiid*StudUn^ p. 88. " If a cone or a cylinder be cut by a pkiae not
parallel to the base» the section b a section of an acute -wigled conc^ which is like a shield

'! CH.n] EUCLID'S OTHER WORKS 17

'( polated version appears in Gregor/s Euclid, and a much earlier and

• f better recension is, says Heiberg\ contained in the MS. Vindobonensis
M philos. Gr. 103, though the end of the treatise, from the middle of

I prop. 16 to the last (18), is missing. The book consists of i8 pro-
positions of j;^A^rrir geometry. Euclid based it on Autolycus' work

y wepl tcipovfiipff^ a^loa^y but also, evidently, on an earlier textbook of
Sphaerka of exclusively mathematical content. It has been con-

I jectured that the latter textbook may have been due to Eudoxus^

* 8. The Optics.

This book needs no description, as it has been edited by Heiberg
recently*, both in its genuine form and in the recension by Theon.
The Cataptrica published by Heiberg in the same volume is not
genuine, and Heiberg suspects that in its present form it may be
Theon 's. Il^is not even certain that Euclid wrote Cataptrica at
all, as Proclus may easily have had Theon's work before him and
inadvertently assigned it to Euclid^.

9. Besides the above-mentioned works, Euclid is said to have
written the Elements of Music* (ai xara fiovaiK^v oTotveiworcA?). Two
treatises are attributed to Euclid in our MSS. of the Musici, the
KaraTOfifj mapovo^, Sectio canonis (the theory of the intervals)*, and the
elaaycaiyif apfLOviKi^ (introduction to harmony). The first, resting on
the Pythagorean theory of music, is mathematical and clearly and well
written, the style and the form of the propositions agreeing well with
what we find in the Elements, Its genuineness is confirmed not only
by internal evidence but by the fact that almost the whole of the
treatise (except the preface) is quoted in extenso, and Euclid is twice
mentioned by name, in the commentary on Ptolemy's Harmonica
published by Wallis and attributed by him to Porphyry, but probably
for the most part compiled by Pappus or some other competent
mathematician'. (On the other hand Tannery set himself to prove
that the treatise is not authentic'.) The second treatise is not Euclid's,
but was written by Cleonides, a pupil of Aristoxenus*.

Lastly, it is worth while to give the Arabians* list of Euclid's
works. I take this from Suter's translation of the list of philosophers
and mathematicians in the Fihrist^ the oldest authority of the kind
that we possess**. "To the writings of Euclid belong further [in
addition to the Elements^ : the book of Phaenomena ; the book of

* Euklui'StudUn^ pp. 50 — I.
' Heiberg, op. cit, p. 46 ; Hultsch, Autolycus^ p. xii ; A. A. Bjornbo, Studien iiber

i Menelaos' Stkdrik (Abhandlungen %ur GtschUhte der mathemcUischen WissenschafUn^ XI v.

1901), p. 56 sqq.
1 • Etulidis opera omnia^ vol. vii. (1895).

* Heiberg, Euclid's Optics^ etc, p. 1. • Proclus, p. 69, 3.

* Published in the Musici Scriptores Graeci^ ed. Jan (Teubner, 1895), pp. 113— 166.
' Jan, Musici Scriptores Gtaeci^ p. 116.

* Comptes rendus de VAcad, des inscriptions et belles-lettres y Paris, 1904, pp. 439^445*
Cf. Bibliotheca Mathemaiicay vi,, 1905-6, p. 325, note i.

* Heiberg, Euklid'Studien, pp. 52—5; Jan, Musici Scriptores Graeci, pp. 169 — 174.
*• H. Suter, Dcu McUhematiker' Verzeichniss im Fihrist in Abhandlungen %ur Geschichte

dir McUkematiky VI., 189a, pp. i — 87 (see especially p. 17). Cf. Casiri, I. 339, 340, and
Garti, pp. 4, 5.

H. E. 2

i8

INTRODUCTION

[CH, II

Given Magnitudes [Data]; the book of Tones, known under the name
of Music, not genuine; the book of Division, emended by Thabit;
the book of Utilisations or Applications [Parisms\ not genuine; the
book of the Canon ; the book of the Heavy and Light ; the book of
Synthesis, not genuine; and the book of Analysis, not genuine/'

It is to be observed that the Arabs already regarded the book of
Tones (by which must be meant the tla^ayoryfi appi^oviKiq) as spurious.
The book of Division is evidently the book on Divisions {of figures).
The next book is described by Casin as " liber de utilitate suppositus."
Sutcr gives reason for believing the Poristns to be meant', but does
not apparently offer any explanation of why the work is supposed to
be spurious. The book of the Canon is clearly the Kara-rop.ri mavov^^i.
The book on *' the Heavy aVid Light '' is apparently the tract t>e itvi
a pondtrosOy included in the Basel Latin translation 0^1537, and in
Gregory's edition. The fragment, however, cannot safely be attributed
to Euclid, for (1) we have nowhere any mention of his having written
on mechanics, (2) it contains the notion of speci6c gravity in a form
so clear that it could hardly be attributed to anyone earlier than
Archimedes'. Suter thinks* that the works on Analysis and Synthesis
(said to be spurious in the extract) may be further developments of
the Data or Parisms^ or may be the interpolated proofs of EucL
Xin. I — ^s» divided into analysis uj\A synt^t^is, as to which sec the notes
on those propositions.

' Suter, <^« Hi, pp. 49, 50. Wenncli trAn&lai^ tb< word as "atUk." Suier i«^s that

the n^rest ttiea^ning of tlie Arabic word u of "porism" is utt^ gain (NuUen, GcwionJ, while
A further meaning is explan anions observation^ addition : a gain arising out of what has
preceded (cT. Proclus' dentition of the porism in the sense of a corollary).
* Heiberg, Euklid-Seudifn, pp. 9, jo» * Suter, ^, cU, p. 50.

CHAPTER III.

GREEK COMMENTATORS ON THE ELEMENTS OTHER
THAN PROCLUS.

That there was no lack of commentaries on the Elements before
the time of Proclus is evident from the terms in which Proclus refers
to them; and he leaves us in equally little doubt as to the value
which, in his opinion, the generality of them possessed. Thus he says
in one place (at the end of his second prologue)^ :

"Before making a beginning with the investigation of details,
I warn those who may read me not to expect from me the things
which have been dinned into our ears ad nauseam (SiaredpvXfjTai) by
those who have preceded me, viz. lemmas, cases, and so forth. For
I am surfeited with these things and shall give little attention to them.
But I shall direct my remarks principally to the points which require
deeper study and contribute to the sum of philosophy, therein emulating
the Pythagoreans who even had this common phrase for what I mean
' a figure and a platform, but not a platform and sixpfrnr***'" —

In another place* he says : " Let us now turn to the elucidation
of the things proved by the writer of the Elements, selecting the more
subtle of the comments made on them by the ancient writers, while
I cutting down their interminable diflfuseness, giving the things which
are more systematic and follow scientific methods, attaching more
importance to the working-out of the real subject-matter than to the
variety of cases and lemmas to which we see recent writers devoting
'.themselves for the most part"

At the end of his commentary on Eucl. I. Proclus remarks* that
^the commentaries then in vogue were full of all sorts of confusion, and
K:ontained no account of causes, no dialectical discrimination, and no
(philosophic thought.

These passages and two others in which Proclus refers to "the

•commentators'" suggest that these commentators were numerous.

He does not however give many names; and no doubt the only

mportant commentaries were those of Heron, Porphyry, and Pappus.

1 Proclus, p. 84, 8.

' i.e. we retxih a certain height, use the platform so attained' as a base on which to build
nother stage, then use that as a base and so on.
* Proclus, p. 100, 10. * idid. p. 433, 15. ' iduf, p. 389, 11 ; p. 338, 16.

2 — 2

M INTRODUCTION [cK. ill

I. Heron*

Proclus alludes to Heron twice as Heron ntechaniois^, in another
place* he associates him with Ctesibius. and in the three other
passages* where Heron is mentioned there is no reason to doubt
that the same person is meant, namely Heron of Alexandria. The
date of Heron is still a vexed question, though the possible limits
appear to have been practically narrowed down to the 150 years
between (say) 50 B,c, and 100 A.U Martin* concluded that Heron
lived till the middle of the first century B.C^ Hultsch* placed him at
the end of the secoqd century B*C, Cantor in his first two editions
took a middle course and gave 100 B.C as the date when he flourished*.
But it is now certain that in his Mec/uinks, preserved in the Arabic
and recently published ^ Heron quotes Posidonius the Stoic (of
Apamea, Cicero's teacher) by name as the author of a definition
of the centre of gravity. Now Posidonius lived till about the middle
of the first century B.C. * and, assuming that his writings dated from
not earlier than 90 or 80 B,C, we must put Heron at all events (say)
fifty years later than Hultsch placed him. Cantor now, while main-
taining that he belong^ to the first century B.C, admits that he may
have flourished as late as the last third of it*.

But in the meantime an entirely different view was elaborated by
W* Schmidt, the editor of the first volume of the new edition of
Heron's complete works, who assigned him to the second half of the
first century a,d.* The arguments for the Urminus post gmm are
mainly these* (i) Vitruvius gives in the preface to Book vii, of his
De ArchiUctura (brought out apparently 14 B.C:) a list of authorities
on machinatiofies from whom he had made extracts. This list contains
twelve names and has every appearance of being scrupulously com-
plete ; but^ while it includes Archytas, Archimedes, Ctesibius, and
Fhilo of Byzantium (who come second, third, fourth, and sixth in
order respectively), Heron is not mentioned. Moreover the pointii
of diflfcrence between Vitruvius and Heron seem on the whole to be
more numerous and important than the resemblances, (2) Diels
concluded from the use of Latinisms by Heron that the first century a»D.
was the earliest possible date. (3) A definite date was derived by
Carra de Vaux from the identification of a small single-screw olive
press described by Heron {Ah^c/mnicSj llh 20) with one mentioned by
Pliny {Nat Hist XViil. 517) as having been introduced within the
last twenty- two years : this gives A.D. 55 as the date before which the
Mechanics could not have been written. The tenninus ante qnetn,
100 A.D., was arrived at (i) from internal evidence suggesting that

^ Proclus, p. 305, 24; p, 34^5, J3»

* iMd. p, 41^ to. ' itid. p. 196, 16; p, 313, 71 p. 419, 13.

* Martin, Reckfri/ui sur la xnt tt Its suvrngn d^Hfr&n d^AU^^mdrity Parish 1854, p* ^7,
' Hultsch, Meir^hgum^m icriptorum reliqumtt 1&64, 1. 9.

* Cantorf Gtuh. d, Math^ i^, p, 347.
' Hsrm^is AUxandrini cpera quae tupertuni pmnia ^Teabner, Leipzig, vol* JL edited b*

L* Nix and W- Schmidt, 1900.

' Cantor, Gtsch. d. Matk. I,, p. 366.

* See ff^rpmi AUxattdnm p/era, vol 1-, 1899, pp, bt— xxv.

■

CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS ax

Heron was earlier than Claudius Ptolemy (about 100-178 A.D.), and
(2) from an apparent reference by Plutarch to a proposition about
incidence and reflexion taking place at equal angles, proved by
Heron in his Catoptricay coupled with the facts that in that work
Heron mentions Menelaus of Alexandria (about 100 AD.) and that
Plutarch died at a great age in 1 20 AD.

Attempts have however been made in two recent tracts to over-
throw almost the whole of these arguments ^ (i) It is asserted that the
olive-press of Mechanics III. 20 is not the same as that referred to by
Pliny. (2) It is pointed out that Heron is mentioned with Archimedes
and Ctesibius in a passage of Proclus which is supposed to be drawn
from Geminus*. But, as Geminus wrote about 70 B.C and Posidonius
not earlier than 90 B.C., while Heron quotes Posidonius and is therefore
later, the intervals are all too short to make it probable that Heron
would be mentioned in Geminus' historical work ; and I think that
the name of Heron may well have been inserted after that of Ctesibius
by Proclus himself. (3) The view that Vitruvius did not use Heron's
work is attacked, and the contrary sought to be proved, on the basis
apparently of three passages, {a) Vitruvius* water-organ is held to
be decidedly better than Heron's* : therefore Vitruvius used Heron's
in order to improve upon it. {b) Vitruvius, in a passage describing
a certain use of the lever, takes a wrong point to be the fulcrum ; and
it is held that he cannot have made the mistake himself, but must
necessarily have copied it from Heron^ In order, however, to find
the same error in Heron, Hoppe arbitrarily alters both the figure and
the text (c) Vitruvius describes the working of a certain crane in
language less clear than that of Heron*; therefore he used Heron but
misunderstood him ! AH would appear to be grist which comes to
the mill of such critics : but I doubt whether such arguments will
convince those who hold to the second half of the first century as the
date that their view is mistaken.

That Heron wrote a systematic commentary on the Elements
might be inferred from Proclus, but it is rendered quite certain by
references to the commentary in Arabian writers, and particularly in
an-Nairizi's commentary on the first ten Books of the Elements. The
Fihrist says, under Euclid, that " Heron wrote a commentary on this
book [the Elements], endeavouring to solve its difficulties*"; and
under Heron, " He wrote : the book of explanation of the obscurities
in Euclid'...." An-Nairlzl's commentary quotes Heron by name very
frequently, and often in such a way as to leave no doubt that the
author had Heron's work actually before him. Thus the extracts are

^ E. Hoppe, Ein Bdtrag tur Zeithatimmung Herons von AUxandrienf Hamburg, 1909 ;
Rudolf Meier, De Heronis atUUe^ Leipzig, 1905. See the references to the arguments in
Cantor, Gesch. d. Math, i„ pp. 365, 367, 545—7.

' Proclus, p. 41, la

' Vitruvius, x. 13 ; Heron, vol. i. p. 19a sqq. (PntuwuUus^ I. 43, 43).

^ Vitruvius, X. 3, 3; Heron, vol. ii. pp. 114 — 116 (Mechanics^ 1 1. 8).

■ Vitruvius, X. 3, 10 ; Heron, vol. ii. pp. 101 — 4 Qifechanus^ ill. a).

* DcLs MaikimoHker- Veneuhniss im Fikrist (tr. Suter), p. 16.

^ ibid, p. as.

» INTRODUCTION [ch, hi

given in the first person introduced by " Heron says *' (" Dixit Yrinus "
or "Heron"); and in other places we are told that Heron '*says
nothing/* or "is not found to have said anything " on such and such
a proposition. The commentary of an-Nair!zi is being published by
Besthorn and Heiberg from a Leiden MS. of the translation of the
Elements by al-Hajjaj with the commentary attached \ But this MS.
only contains six Books, and several pages in the first Book are
missing> which contain the comments of Simphcius on the first twenty-
two definitions of the first Book, Fortunately the commentary of
an-Nairizi has been discovered in a more complete form, in a Latin
translation by Gherardus Cremonensis of the twelfth century, which
contains the missing comments by Simplicius and an-Nairizl's com-
ments on the first ten Books. This valuable work has recently been
edited by Curtzel

Thus from the three sources, Prod us, and the two versions of
an-NairlzI, which supplement one another, we are able to form a very
good idea of the character of Heron's commentary. In some cases
observations given by Proclus without the name of their author are
seen from an-Nairlzl to be Heron's ; in a few cases notes attributed
by Proclus to Heron are found in an-Nairizi without Heron's name ;
and J curiously enough, one alternative proof (of L 25) given as Heron's
by Proclus is introduced by the Arab with the remark that he has
not been able to discover who is the author.

Speaking generally, the comments of Heron do not seem to have
contained much that can be called important We find

(i) A few general notes, eg. that Heron would not admit more
than three axioms.

(2) Distinctions of a number of particular cases of Euclid's pro-
positions according as the figure is drawn in one way or in another.

Of this class are the different cases of L 35, 36, III, 7, 8 (where the
chords to be compared are drawn on different sides of the diameter
instead of on the same side), III. 12 (which is not Euclid's, but Heron's
own, adding the case of external contact to that of internal contact in
III, 11), VL 19 (where the triangle in which an additional line is drawn
is taken to be the smaller of the two), vii, 19 (where he gives the
particular case of three numbers in continued proportion^ instead of
four proportionals).

(3) Alternative proofs. Of these there should be mentioned {a)
the proofs of II. i — 10 " without a figure/' being simply the algebraic
forms of proof, easy but uninstructive, which are so popular nowadays^
the proof of 11 1, 25 (placed after 11 L 30 and starting from the arc
instead of the chord), IIL 10 (proved by llh 9), m, 13 (a proof
preceded by a lemma to the effect that a straight line cannot meet a
circle in more than two points). Another class of alternative proof is

^ CodAX Lddtmis 399, t, Euclidis Ekmmia ex iHt£rpr^iUi<mt ai-Hadstkdickndschii
cum amrturUariii ai-Naruii. Two parts carrying the work to the end of Book I, were
tssQed in 189^ and 1897 lespectively' Another part came out in 1905.

* Anariitt in decern iibros £rwrts iiemeniarum Euciidu t&mmeniarii tx mUfprd^xUone
Ghm^rdi Cremonensis.., aUdU Maxuniliaous Curtze (Teubner, Lelpdg^ 'S99)<

I

n

CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS 23

{b) that which is intended to meet a particular objection (h^tmuri^)
which had been or might be raised to Euclid's construction. Thus
in certain cases he avoids producing a particular straight line, where
Euclid produces it, in order to meet the objection of any one who should
deny our right to assume that there is any space available\ Of this
class are Heron's proofs of I. 1 1, 1. 20, and his note on I. 16. Similarly
on I. 48 he supposes the right-angled triangle which is constructed to
be constructed on the same side of the common side as the given
triangle is. A third class (c) is that which avoids reductio ad
absurdum. Thus, instead of indirect proofs. Heron gives direct

K roofs of I. 19 (for which he requires, and gives, a preliminary
jmma), and of i. 25.

(4) Heron supplies certain converses of Euclid's propositions,
e.g. converses of li. 12, 13, vill. 27.

(5) A few additions to, and extensions of, Euclid's propositions
are also found. Some are unimportant, e.g. the construction of isosceles
and scalene triangles in a note on I. i, the construction of two tangents
in III. 17, the remark that Vll. 3 about finding the greatest common
measure of three numbers can be applied to as many numbers as we
please (as Euclid tacitly assumes in VII. 31). The most important
extension is that of ill. 20 to the case where the angle at the
circumference is greater than a right angle, and the direct deduction
from this extension of the result of III. 22. Interesting also are the
notes on I. 37 (on I. 24 in Proclus), where Heron proves that two
triangles with two sides of one equal to two sides of the other and
with the included angles supplementary are equal, and compares the
areas where the sum of the two included angles (one being supposed
greater than the other) is less or greater than two right angles, and
on I. 47, where there is a proof (depending on preliminary lemmas) of
the fact that, in the figure of the proposition, the straight lines AL^
BKy CF meet in a point. After iv. 16 there is a proof that, in a
regular polygon with an even number of sides, the bisector of one
angle also bisects its opposite, and an enunciation of the corresponding
proposition for a regular polygon with an odd number of sides.

Van Pesch" gives reason for attributing to Heron certain other
notes found in Proclus, viz. that they are designed to meet the same
sort of points as Heron had in view jn other notes undoubtedly written
by him. These are (a) alternative proofs of I. 5, I. 17, and l. 32,
which avoid the producing of certain straight lines, {p) an alternative
proof of I. 9 avoiding the construction of the equilateral triangle on
the side of BC opposite to A ; (^r) partial converses of I. 35 — 38, starting
from the equality of the areas and the fact of the parallelograms or
triangles being in the same parallels, and proving that the bases are
the same or equal, may also be Heron's. Van Pesch further supposes
that it was in Heron's commentary that the proof by Menelaus of
L 25 and the proof by Philo of i. 8 were given.

^ Cf. Prodns, 375, 7 c/ M X^oc rif rbww fi^ e^d^cu..., 389, 18 X^ci oSy nt 5rc oix Irrc
^ . rinrot.,.,
c * De ProclifoniibuSt Lugduni-Batavorum, 1900.

11

\l

24 INTRODUCTION [ch< hi

The last reference to Heron made by an-Nairt^I occurs in the note
on VI IL 27, so that the commentary of the former must at least have
reached that point.

II* Porphyry.

The Porphyry here mentioned is of course the Neo-PIatonist who
lived about 232-304 A.D. Whether he really wrote a systematic
commentary on the £/anmfs is uncertain. The passages in Froclus
which seem to make this probable are two in which he mentions him
(i) as having demonstrated the necessity of the words "not on the
same side" in the enunciation of L 14\ and (2) as having pointed out
the necessity of understanding correctly the enunciation of L 26, since,
if the particular injunctions as to the sides of the triangles to be taken
as equal are not regarded, the student may easily fall into error'. ,
These passages, showing that Porphyry carefully analysed Euclid *s
enunciations in these cases, certainly suggest that his remarks were
part of a systematic commentary. Further, the list of mathematicians
in the Fihrist gives Porphyry as having written '*a book on the
Elements," It is true that Wenrich takes this book to have been a
work by Porphyry mentioned by Suidas and Proclus ( Tfuohg. Platen, \
TT^pi apymv libri IL*

There is nothing of importance in the notes attributed to Potphyry
by Proclus.

(i) Three alternative proofs of L 20, which avoid producing a side
of the triangle, are assigned to Heron and Porphyry without saying
which belonged to which. If the first of the three was Heron's, I
agree with van Pesch that it is more probable that the two others
were both Porphyry's than that the second was Heron's and only the
third Porphyry's. For they are similar in character, and the third
uses a result obtained in the second*.

(2) Porphyry gave an alternative proof of L iS to meet a childish
objection which is supposed to require the part of ^C equal to AB to
be cut off from CA and not from A C.

Proclus gives a pr^iisely similar alternative proof of I, 6 to meet a
similar supposed objection ; and it may well be that, though Proclus
mentions no name, this proof was also Porphyry's, as van P^ch
suggests*.

Two other references to Porpbyry found in Proclus cannot have
anything to do with commentaries on the Elements. In the first a
work called the Xu^fiiKTa is quoted, while in the second a philo-
sophical question is raised,

in. Pappus.

The references to Pappus in Proclus arc not numerous ; but we
have other evidence that he wrote a commentary on the Elements,
Thus a scholiast on the definitions of the Data uses the phrase " as

* Proclas, pp. 197, t— 1981 10. * iMd. p. 353, 13, 14 and ihe pages prtcedjng.

* Fihrist (tr, Suter), p. 9^ lo and p> 45 {note j).

* Van Pesch, Di Pr&cH fantihu^ pp. 119, 130* Heibcrg assigned Ibero as above in hts
Eukiid-Studun (p, t6o)t but «eems to OAve changed Ms view later. (See Be$thoni*Heibetg,
C^tM LiidfHjis^ p- 93, note 3.)

* Van Pesch, ^* fit, pp. 13^— j. _ ^

l^

CH. Ill] GREEK COMMENTATORS OTHER THAN PROCLUS 25

Pappus says at the b^inning of his (commentary) on the loth (book)
of Euclid*." Again in the Fikrist we are told that Pappus wrote a
commentary to the tenth book of Euclid in two parts*. Fragments
of this still survive in a MS. described by Woepcke*, Paris. No. 952. 2
(supplement arabe de la Biblioth^que imp^riale), which contains a
translation by Abu 'Uthman (beginning of loth century) of a Greek
commentary on Book x. It is in two books, and there can now be
no doubt that the author of the Greek commentary was Pappus*.
Again Eutocius, in his note on Archimedes, On the Sphere and
Cylinder!, 13, says that Pappus explained in his commentary on the
Elements how to inscribe in a circle a polygon similar to a polygon
inscribed in another circle ; and this would presumably come in his
commentary on Book xil., just as the problem is solved in the second
scholium on Eucl. xil. i. Thus Pappus' commentary on the Elements
must have been pretty complete, an additional confirmation of this
supposition being forthcoming in the reference of Marinus (a pupil
and follower of Proclus) in his preface to the Data to "the com-
mentaries of Pappus on the book'."

The actual references to Pappus in Proclus are as follows:
(i) On the Postulate (4) that all right angles are equal, Pappus is
quoted as saying that the converse, viz. that all angles equal to a
right angle are right, is not true*, since the angle included between
the arcs of two semicircles which are equal, and have their diameters
at right angles and terminating at one point, is equal to a right angle,
but is not a right angle.

(2) On the axioms Pappus is quoted as saying that, in addition to
Euclid's axioms, others are on record as well (avpavaypaf^ecrOai) about
unequals added to equals and equals added to unequals^; these, says
Proclus, follow from the Euclidean axioms, while* others given by
Pappus are involved by the definitions, namely those which assert
that *' all parts of the plane and of the straight line coincide with one
another," that " a point divides a straight line, a line a surface, and a
surface a solid/' and that "the infinite is (obtained) in magnitudes
both by addition and diminution'."

^ Euclid's Da/a, ed. Menge, p. 363. ' Fikrist (tr. Suter), p. 29.

• Mhnoires prheniis h tacacUmie des sciences^ 1856, XI v. pp. 658 — 710.

* Woepcke read the name of the aathor, in the title of the first book zsB .los (the dot
representing a missing vowel). He quotes tdso from other MSS. (e.g. of the Ta'riJkh ai-
aukamU and of the Fikrist) where he reads the name of the commentator %& B ,lis^ B,n.s
OT B./,s. Woepcke takes this author to be Vtdens, and thinks it possible that he may be
the same as the astrologer Vettius Valens. This Heiberg {Euk/ia-Stsuiien, pp. 169, i7o)
proves to be impossible, because, while one of the Mss. quoted by Woepcke says that
**B,M.s, le Roimi'' (late-Greek) was later than Claudius Ptolemy and the Fikrist says
**B./,s, \e RoUmi" wrote a commentary on Ptolemy's Plcmispkaerium, Vettius Valens
seems to have lived under Hadrian, and must therefore have been an eider contemporary of
Ptolemv. But Suter shows (Fikrist^ p. 22 and p. 54, note 9)) that Bancs is only distin-
guished from Badcs by the position of a certain dot, and Ba/os may also easily have arisen
from an original Bados (there is no P in Arabic), so that Pappus must be the person meant.
This is further confirmed by the fact that the Fikrist gives this author and Valens as the
subjects of two separate paragraphs, attributing to the latter astrological works only.

" Heiberg, Euklid-Studien^ p. 173; Euclid's Daia^ ed. Mexige, pp. 356, Hi.

* Proclus, pp. 189, 190. ' ibid. p. 197, (^10.

• ihid, p. 198, 3—15.

26 INTRODUCTION

(3) Pappus gave a pretty proof of L 5. This proof has, I think;
been wrongly understood ; on this point sec my note on the
proposition.

(4) On h 47 Proclus says^ : ** As the proof of the writer of the
Elements is manifest, I think that it is not necessary to add anything
further, but that what has been said is sufficient, since indeed those
who have added more^ like Heron and Pappus, were obliged to make
use of what is proved in the sixth book, without attaining any
important result.'* We shall see what Heron's addition consisted of;
what Pappus may have added we do not know, unless it was some-
thing on the lines of his extension of L 47 found in the Syfu^ge
(iv/p. 176, ed. Hultsch).

We may fairly conclude, with van Pesch*, that Pappus is drawn
upon in various other passages of Proclus where he quotes no
authority, but where the subject-matter reminds us of other notes
expressly assigned to Pappus or of what we otherwise know to have
been favourite questions with him. Thus :

r We are reminded of the curvilineal angle which is equal to but
not a right angle by the note on I. 32 to the efllcct that the converse
(that a figure with its interior angles together equal to two right
angles is a triangle) is not true unless we confine ourselves to
rectilineal figures. This statement is supported by reference to a
figure formed by four semicircles whose diameters form a square, and
one of which is turned inwards while the others are turned outwards.
The figure forms two angles "equal to" right angles in the sense
described by Pappus on Post 4, while the other curvilineal angles are
not considered to be angles at all, and are left out in summing the
internal angles. Similarly the allusions in the notes on L 4, 23 to
curvilineal angles of which certain moon-shaped angles {p.i}PQ€i&€h)
are shown to be "equal to*' rectilineal angles savour of Pappus,

2, On L 9 Proclus says* that ** Others, starting from the Archi-
medean spirals, divided any given rectilineal angle in any given ratio."
We cannot but compare this with Pappus IV. p, 286, where the spiral
is so used ; hence this note, including remarks immediately preceding
about the conchoid and the quadratrix, which were used for the same
purpose, may very well be due to Pappus.

3, The subject of isopen metric figures was a favourite one with
Pappus, who wrote a recension of Zenodorus' treatise on the subject*
Now on L 35 Proclus speaks* about the paradox of parallelograms
having equal area (between the same parallels) though the two sides
between the parallels may be of any length, adding that of parallelo-
grams with equal perimeter the rectangle is greatest if the base be
given, and the square greatest if the base be not given etc. He
returns to the subject on I. 37 about triangles* Compare' also his
note on l. 4. These notes may have been taken from Pappus.

* Proclus p, 419, g— '5-

' Van Pcsch, Dt Procli f^rUibtt^^ y, 134 sqq. * Proclus, p. 171, jo*

* Pappu^ V. pp. 304—^50; for Zenodorus own treatise see HuJ tech's Appendix, pp. it 89
— i*n*

* Produ% pp. 396 — 8. • iMd. pp. 403 — 4- '' iHd. pp. 336 — 7, t

CH-Jii] GREEK COMMENTATORS OTHER THAN PROCLUS a?

4. Again, on I. 21, Proclus remarks on the paradox that straight
lines may be drawn from the base to a point within a triangle which
are (i) together greater than the two sides, and (2) include a less
angle provided that the straight lines may be drawn from points in
the base other than its extremities. The subject of straight lines
satisfying condition (i) was treated at length, with reference to a
variety of cases, by Pappus*, after a collection of "paradoxes" by
Erycinus, of whom nothing more is known. Proclus gives Pappus'
first case, and adds a rather useless proof of the possibility of drawing
straight lines satisfying condition (2) alone, adding that " the proposi-
tion stated has been proved by me without using the parallels of
the commentators'." By " the commentators " Pappus is doubtless
meant

5. Lastly, the "four-sided triangle," called by Zenodorus the
••hollow-angled,"* is mentioned in the notes on I. Def. 24 — 29 and
I. 21. As Pappus wrote on Zenodorus' work in which the term
occurred^ Pappus may be responsible for these notes.

IV. Simplicius.

According to the Fihrist^, Simplicius the Greek wrote ''a com-
mentary to the beginning of Euclid's book, which forms an introduc-
tion to geometry." And in fact this commentary on the definitions,
postulates and axioms (including the postulate known as the Parallel-
Axiom) is preserved in the Arabic commentary of an-Nairizi*. On
two subjects this commentary of Simplicius quotes a certain " Aganis,"
the first subject being the definition of an angle, and the second the
definition of parallels and the parallel-postulate. Simplicius gives
word for word, in a long passage placed by an-NairlzI after I. 29, an
attempt by •• Aganis " to prove the parallel-postulate. It starts from
a definition of parallels which agrees with Geminus' view of them as
given by Proclus^ and is closely connected with the definition given
by Posidonius*. Hence it has been assumed that "Aganis" is none
other than Geminus, and the historical importance of the commentary
of Simplicius has been judged accordingly. But it has been recently
shown by Tannery that the identification of " Aganis " with Geminus
is practically impossible*. In the translation of Besthom-Heiberg
Aganis is called by Simplicius in one place " philosophus Aganis," in
another " magister noster Aganis," in Gherard's version he is " socius
Aganis" and "socius noster Aganis." These expressions seem to
leave no doubt that Aganis was a contemporary and friend, if not
master, of Simplicius ; and it is impossible to suppose that Simplicius
(fl. about 500 A.D.) could have used them of a man who lived four and

• Pappus III. jM). 104—130. » Proclus, p. 318, 15.

' Proclus, p. 105, 14; cf. pp. 318, 339. * See Pappus, ed. Hultsch, pp. 11 54, 1306.

• Fihrist (tr. Suter), p. ii.

• An-NairizI, cd. Bestbom-Heiberg, pp. 9—41, 119;— 1331 cd. Curtze, pp. i — 37, 65 — 73.
The Codix LHdensiSy from which ^sthom and Heiberg are editing the work, has un-
fortunately lost some leaves so that there is a gap from Def. i to Def. 35 (parallels). The
loss is, however, made good by Curtze's edition of the translation by Ghenmi of Cremona.

' Proclus, p. 177, 31. • ibid, p. 176, 7.

' Biblioikeca Mathematical ii^, 1900, pp. 9 — 11.

"1

tS INTRODUCTION [ciL m J

a half centuries before his time. A phrase in Simplicius* word-for-
word quotation from Aganis leads to the same conclusion. He speaks
of people who objected *'cven in ancient times" (iam antiquitus) to ^
the use by geometers of this postulate. This would not have been an |

appropriate phrase had Gcminus been the writer I do not think
that this difficulty can be got over by Suter's suggestion* that the
passages in question may have been taken out of Heron's commentary,
and that an-Nairizi may have forgotten to name the author ; it seems
clear that Simplicius is the person who described " Aganis." Hence
we are driven to suppose that Aganis was not Geminus, but some
unknown contemporary of Simplicius*, Considerable interest will
however continue to attach to the comments of Simplicius so
fortunately preserved*

Proclus tells us that one Aegaeas {? Aenaeas) of Hierapolis wrote an
epitome of the Ekments^ ; but we know nothing more of him or of it

1 Z€its€krififur Ai^th. u. Pkysik, XLIV., hLit.-tttt Abtb Jf. 6j*

^ The ibove Aripimeni seems Lome quUc infiuperahte. Ine other Ai^inents of Tvunery

do Dot* however, carry conviction to my mind* I dci ttot follow the reaM>niiig bailed on
Aganis' definition of an angle. It appears to me a pure assumption ihat Geminus would have
seen that Po^idonius^ definition of pftralleb w^ not admissible. Nor 6ot^ it seem to me to
count for mych that Proclus, while telling us that Geminus held that the postulate ought to be
proved and warned the unwaiy again^ hastUy concluding that two straight lines approaching
one another mu&t necci^sarity meet (cf« a curve and it& asymptote), gives no nint that
Geminus did try to prove the postulate^ It may well be tnat Procltis omitted Getninns*
"proofs (if he trrote one) because he preferred Ptolemy*s attempt which he give

(PP- 3^5—7)'

* Proclus, p. 361* 11*

CHAPTER IV.

PROCLUS AND HIS SOURCES^.

It is well known that the commentary of Proclus on Eucl. Book I.
is one of the two main sources of information as to the history of
Greek geometry which we possess, the other being the Collection of
Pappus. They are the more precious because the original works of
the forerunners of Euclid, Archimedes and Apollonius are lost, having
probably been discarded and forgotten almost immediately after the
appearance of the masterpieces of that great trio.

Proclus himself lived 410-485 A.D., so that there had already
passed a sufficient amount of time for the tradition relating to the
pre-Euclidean geometers to become obscure and defective. In this
connexion a passage is quoted from Simplicius' who, in his account
of the quadrature of certain lunes by Hippocrates of Chios, while
mentioning two authorities for his statements, Alexander Aphro-
disiensis (about 220 A.D.) and Eudemus, says in one place', ''As
regards Hippocrates of Chios we must pay more attention to Eudemus,
sittce he was nearer the times, being a pupil of Aristotle."

The importance therefore of a critical examination of Proclus'
commentary with a view to determining from what original sources
he drew need not be further emphasised.

Proclus received his early training in Alexandria, where Olympio-
dorus was his instructor in the works of Aristotle, and mathematics
was taught him by one Heron* (of course a different Heron from the
'^mechanicus Hero" of whom we have already spoken). He after-
wards went to Athens where he was imbued by Plutarch, and by
Syrianus, with the Neo-Platonic philosophy, to which he then devoted

^ My task in this chapter is made easy by the appearance, in the nick of time, of the
dissertation De Prodi fotUitms by J. G. van Pesch (Lugduni-Batavorum, Apud L. van
Nifterik, MDCCCC). The chapters dealing directly with the subject show a thorough
acquaintance on the part of the author with all the literature bearing on it; he covers
the whole field and he exercises a sound and sober judgment in forming his conclusions.
The same cannot always be said of his only predecessor in the same inquiry, Tannery
(in La GhmHrie grtcque, 1887), ^^^ often robs his speculations of much of their value
through his proneness to run away with an idea ; he does so in this case, basing most of his
conclusions on an arbitrary and unwarranted assumption as to the significance of the words
ol repf rua (e.g.'Upfura, no^ctdt&rtov etc) as used in Proclus.

• Simplicius on Aristotle*s Physics^ ed. Dieb, pp. 54 — 69.

* ibid, p. 6%, 31.

^ Cf. Martin, Recktrcha sur lavidet Us omfrages d^Hiron dAUxandrii, pp. 340 — 3.

30 INTRODUCTION [ch. iv

heart and soul, becoming one of its most prominent exponents* He
speaks everywhere with the highest respect of his masters, and
was in turn regarded with extravagant veneration by his contem-
poraries, as we learn from Marinus his pupil and biographer. On
the death of Syrian us he was put at the head of the Neo-Platonic
school He was a man of untiring industry^ as is shown by the
number of books which he WTote, including a large number of com-
mentaries, mostly on the dialogues of Plato* He was an acute
dialectician, and pre- eminent among his contemporaries in the
range of his learning^; he was a competent mathematician; he was
even a poet At the same time he was a believer in all sorts of
myths and mysteries and a devout worshipper of divinities both
Greek and Oriental,

Though he was a competent mathematician, he was evidently
4nuch more a philosopher than a mathematician I This is shown
even in his commentary on EucL !.» where, not only in the Prologues
(especially the first^ but also in the notes themselves, he seizes any
opportunity for a philosophical digression. He says himself that he
attaches most importance to **the things which require deeper study
and contribute to the sum of philosophy*"; alternative proofs, cases ^
and the like (though he gives many) have no attraction for him ;
and, in particular, he attaches no value to the addition of Heron to
L 47*, which is of considerable mathematical interest Though he
esteemed mathematics highly, it was only as a handmaid to philosophy.
He quotes Plato's opinion to the effect that "mathematics, as making
use of hypotheses, falls short of the non-hypothetical and perfect
science*". »/' Let us then not say that Plato excludes mathematics
from the sciences, but that he declares it to be secondary to the one
supreme science V And again, while "mathematical science must be
considered desirable in itself, though not with reference to the needs
of daily lifej'' **if it is necessary to refer the benefit arising from it to
, something else, we must connect that benefit with intellectual know-
ledge {voepcLv yptii>a-tp)y to which it leads the way and is a propaedeutic,
clearing the eye of the soul and takii)g away the impediments which
the senses place in the way of the knowledge of universals {twv
Skmvyr

We know that in the Neo* PI atonic school the younger pupils
learnt mathematics ; and it is clear that Proclus taught this subject,
and that this was the origin of the commentary. Many passages
show him as a master speaking to scholars. Thus "we have illustrated

^ Edict c^ Is him '* Der GelchrtCi dem kein Feld damaligen Wissens verschlos^cn ist/^
' Van Tesch observes that in his commentaries on the Tiwa^iw (pp* 67;— 1> he fipe^ks
as no real maihemaiician could h^ve spoken. In the passage referred Lo ihe qu^uon h
whether the &un occupies a middie place among the planets^ Proclus rejects the view of
Hipparchus and Ptolemy because **d &toupy6v^^ ($c. the Chaldean, says Zellcr) thinks otherwise ,
"whom it is not lawful to disbeJieve/' Martin says rather neatly^ *^ Pour ProduSi les
Elements d'EucUde ont Thcnrcu^e chance de n'^tre contr«dits ni par fei Omclcs cbaldaSfques,
tu par tes speculations des pythtigoneicns ancicns et nouveaux......"

• Proclus, p* S41 13* ' iHd. p, 419^ Up

* ibid^ p. 3t, 30» • ibid, p. %t, %" i
^ itid. p. a;, a; to aS, 7; cf. alio p. 11, 95, pp» 46, 47.

CH. TV] PROCLUS AND HIS SOURCES 31

and made plain all these things in the case of the first problem, but
it is necessary that my hearers should make the same inquiry as
regards the others as welP," and '' I do not indicate these things as a
merely incidental matter but as preparing us beforehand for the
doctrine of the Timaeusl" Further, the pupils whom he was
I • addressing were beginners in mathematics ; for in one place he says
^ that he omits " for the present" to speak of the discoveries of those
I who employed the curves of Nicomedes and Hippias for trisecting
I an angle, and of those who used the Archimedean spiral for dividing
an angle in any given ratio, because these things would be too
difficult for beginners (SvaOeapiJTovi roU elaayofiepoi^y. Again, if
his pupils had not been beginners, it would not have been necessary
for Proclus to explain what is meant by saying that sides subtend
certain angles^ the difference between adjacent and vertical angles'
etc., or to exhort them, as he often does, to work out other particular
cases for themselves, for practice {yviiva^ria^ ev€Ka)\

The commentary seems then to have been founded on Proclus'
lectures to beginners in mathematics. But there are signs that it
was revised and re-edited for a larger public ; thus he gives notice in
one place' "to those who shall come upon" his work (to*9 ivrev^o-
fihoi^y There are also passages which could not have been under-
stood by the beginners to whom he lectured, e.g. passages about the
cylindrical helix', conchoids and cissoids'. These passages may have
been added in the revised edition, or, as van Pesch conjectures, the
explanations given in the lectures may have been much fuller and
more comprehensible to beginners, and they may have been shortened
on revision.

In his comments on the propositions of Euclid, Proclus generally
proceeds in this way : first he gives explanations r^arding Euclid's
proofs, secondly he gives a few different cases, mainly for the sake of
practice, and thirdly he addresses himself to refuting objections
raised by cavillers to particular propositions. The latter class of
note he deems necessary because of "sophistical cavils" and the
attitude of the people who rejoiced in finding paralogisms and in
causing annoyance to scientific men". His commentary does not
seem to have been written for the purpose of correcting or improving
Euclid. For there are very few passages of mathematical content
in which Proclus can be supposed to be propounding anythirtg of his
own ; nearly all are taken from the works of others, mostly earlier
commentators, so that, for the purpose of improving on or correcting
Euclid, there was no need for his commentary at all. Indeed only in
one place does he definitely bring forward anything of his own to get
over a difficulty which he finds in Euclid"; this is where he tries to

• Proclus, p. a 10, 18. • iMd, p. 384, 1.

• Mi, p. 171, II. * Md. p. 138, 13.

• Md. p. 198, 14. • Cf. p. 314, 15 (on 1. 1).
' iHd. p. 84, g. • idtd, p. 105.

9 .•A.W ^ .7^ 10 .A.V/ n -.fe ft

wvut, ^* 04, y*

' Md. p. 113.

" Md. pp. 368—373.

*• lAfV/. p. 375, 8.

i-

3a INTRODUCTION [ch. iv

prove the parallel-postulate, after first giving Ptolemy's attempt and
then pointing out objections to it. On the other hand there are a
number of passages in which he extols Euclid ; thrice* also he
supports Euclid against Apollonius where the laUer had given proofs
which he considered better than Euclid's (I. lo, ii, and 23).

Allusion must be made to the debated question whether Proclus
continued his commentaries beyond Book L His intention to do so
is clear from the following passages. Just after the words above
quQted about the trisection etc* of an angle by means of certain curves
he says, '* For we may perhaps more appropriately examine these
things on the third book, where the writer of the Elements bisects a
given circumference^" Again, after saying that of all paraJlelograms
which have the same perimeter the square is the greatest *' and the
rhomboid least of all/' he adds: "But this we wilt prove in another
place; for it is more appropriate to the (discussion of the) hypotheses
of the second book* " Lastly, when alluding (on h 45) to the squaring
of the circle, and to Archimedes' proposition that any circle is equal
to the right-angled triangle in which the perpendicular is equal to the
radius of the circle and the base to its perimeter, he adds, '' But of this
elsewhere**'; this may imply an intention to treat of the subject on
Eucl. XIL, though Heiberg doubts it** But it is clear that, at the time
when the commentary on Book I. was written* Proclus had not yet
b^^n to write on the other Books and was uncertain whether he
would be able to do so : for at the end he says\ '* For my part, if I
should be able to discuss the other books^ in the same manner, I
should give thanks to the gods ; but, if other cares should draw me
away, I beg those who are attracted by this subject to complete the
exposition of the other books as well, following the same method, and
addressing themselves throughout to the deeper and better denned
questions involved '' (to 'Trpayfjt^aretwS^^ vapra^ov xal ev&iaiperov
fLeraStoi Kovras:)-

There is in fact no satisfactory evidence that Proclus did actually
write any more commentaries than that on Book J*, those who have
attributed to Proclus some of the scholia on the later books having
failed to prove their case\ The contrary view receives support from
two facts pointed out by Heiberg, viz. (i) that the scholiast's copy
of Proclus was not much better than our MSS. : in particular, it had

* Proclus, p. aSo, gj p. 183* 10; pp, 335, $36, * i6id. p» 371, 14*
■ i^J. p. J98, 1 8, * ihui, p, 433, 6.

• Heiberg J £uklid*Studi£n, p* 165, note. ' Proclus, p. +33* 9.

^ The words in ihe Greek arc: ti ptHv Sifwi^Btlnifiet x^i roTt XiHiro7» m ovr^ rpi/wof
^\$w. For i^t\9ttif Heiberg would read iwf^fMtty.

' Heiberg {EuMUd-Sindkn, p- 166) gives reason for doubling the evidoice mdduoed
by Waclumutht by which Knoche vras persuaded to give up his ori^nal view (ha.t Proclus did
nai write any more comtnentaries. Wacbimutb tie lies sole-ly upon a Vatican MS- which ha>i
at the head of a collection of scholia on Books K (extracts from the e sclent convmcnUry
of Proclus), IL, Vm VI., 9C. the title ; Eti ri EujfXff^n; ^rwx^ia wpo\aft^iu/6fiiPa in tQv Iip6ii\o*t
€wopdhfw jfol jcat' iwiT^^. Hcibeig holds Umt this liile its^Jf makes ti probable that the
anthor^ip ascribed to Proclus was restricted to the scholia on Book J.: otherwise how
could one understand the expression irpoXati^ti^btitwa in tmp llp6K\ov^ which words would
suit extracts ^m Frndtu'/fW^s^tfi weU enough, but not the acholia to latei^ Books?

CH. nr] PROCLUS AND HIS SOURCES 33

the same lacunae in the notes on l. 36, 37 and 1. 41 — ^43 : and even this
fact makes it improbable that the scholiast had further commentaries
of Proclus which have vanished for us ; (2) that there is no trace
in the scholia of the notes which Proclus promised in the passages
quoted above \

Coming now to the question of the sources of Proclus, we may say
that everything goes to show that his commentary is a compilation,
though a compilation "in the better sense" of the term*. He does
not even give us to understand that we shall find in it much of his own ;
" let us," he says, " now turn to the exposition of the theorems proved
by Euclid, selecting the more subtle of the comments made on them
by the ancient writers, and cutting down their interminable difluse-
ness...*'': not a word about anything of his own. At the same time,
he seems to imply that he will not necessarily on each occasion quote
the source of each extract from an earlier commentary ; and, in fact,
while he quotes the name of his authority in many places, especially
where the subject is important, in many others, where it is equally
certain that he is not giving anything of his own, he mentions no
authority. Thus he quotes Heron by name six times ; but we now
know, from the commentary of an-NairlzI, that a number of other
passages, where he mentions no name, are taken from Heron, and
among them the not unimportant addition of an alternative proof to
I. 19. Hence we can by no means conclude that, where no authority
is mentioned, Proclus is giving notes of his own. The presumption is
generally the other way ; and it is often possible to arrive at a con-
clusion, either that a particular note is not Proclus' own, or that it
is definitely attributable to someone else, by applying the ordinary
principles of criticism. Thus, where the note shows an unmistakable
affinity to another which Proclus definitely attributes to some com-
mentator by name, especially when both contain some peculiar and
distinctive idea, we cannot have much doubt in assigning both to the
same commentator^ Again, van Pesch finds a criterion in the form
of a note, where the explanation is so condensed as to be only just
intelligible ; the note is that in which a converse of I. 32 is proved*,
the proposition namely that a rectilineal figure which has all its
interior angles together equal to two right angles is a triangle.

It is not safe to attribute a passage to Proclus himself because he
uses the first person in such expressions as " I say " or " I will prove '*
— for he was in the habit of putting into his own words the substance
of notes borrowed from others — nor because, in speaking of an

1 Heiberg, Euklid'Studim, pp. 167, 168.

' Kooche, UfUersuchungen iiber des Froklus Diadochus Comnuntar tu Euklitts EU-
menien (1863) p. 11.

■ Proclus, p. aoo, 10 — 15.

* Instances of the application of this criterion will be found in the discussion of Proclus*
indebtedness to the commentaries of Heron, Porphyry and Pappus.

* Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the
proof is followed by a passage which, on comparison with Pappu? note on the postulate that
all right angles are equal, he feeb justified in assigning to Pappus. I doubt if ttie evidence is
sufficient.

H. K. 3

U INTRODUCTION [cH. nr

objection raised to a particular proposition, he uses such expressions
as "perhaps someone may object" (fir<w? S* av rtj^cT ipirraUp^^*): for
sometimes other words in the same passage indicate that the objection
had actually been taken by someone'. Speaking generally, we shall
not be justified in concluding that Proclus is stating something new of
his own unless he indicates this himself in express terms.

As regards the form of Proclus* references to others by name, van
Pesch notes that he very seldom mentions the particular wark from
which he is borrowing. If we leave out of account the references to
Plato's dialogues, there are only the following references to books :
tiieBac^ia^ of Philolaus\ the Symmikta of Porphyry", Archimedes Oh
the Sphere and Cyiindtr^^ Apollonius On the cochiias^^ a book by
Eudemus on The Ang'ie\ a whole book of Posidonius directed against
Zeno of the Epicurean sect^ Carpus* Astrofwmy^, Eudemus' History of
Geometry^ ^ and a tract by Ptolemy on the parallel -postulate".

Again, Proclus does not always indicate that he is quoting some-
thing at second-hand. He often does so, e.g. he quotes Heron as the
authority for a statement about PhiJippus, Eudemus as attributing a
certain theorem to Oenopides etc.; but he says on h \2 that "Oeno-
pides first investigated this problem, thinking it useful for astronomy** .
when he cannot have had Oenopides* work before him. i

It has been said above that Proclus was in the habit of stating in *
his own words the substance of the things which he borrowed. We j
are prepared for this when we find him stating that he will select the
best things from ancient commentaries and "cut short their intermin-
able diffuseness/* that he will * briefiy describe '* {avvrofi^K; itrTopTJaai)
the other proofs of I. 20 given by Heron and Porphyry and also the
proofs of I, 25 by M^nelaus and Heron, But the best evidence is of
course to be found in the passages where he quotes works still extant,
e*g. those of Plato, Aristotle and Plotinus, Examination of these
passages shows great divergences from the original ; even where he
purports to quote textual ly, using the expressions *' Plato saySj'' or
•' Plotinus says," he by no means quotes word for word". In fact, he
seems to have had a positive distaste for quoting textually from other
works. He cannot conquer this even when quoting from Euclid ; he
says in his note on I. 22, "we will follow the words of the geometer*'
but fails, nevertheless, to reproduce the text of Euclid unchanged".

We now come to the sources themselves from which Proclus drew

^ Van Pesch illustrates this by an objection refuted in the note on i. 9, p. 173, ir sqq.
After using the above expression to introduce the objection, Proclus uses further on (p. 373, 15)
the term **they say" {^aaU^).

» Proclus, p. 11, 15. ' idiit. p. 56, 15.

* Mt.p. 71, 18. • tiiJ, p. J05, 5.

* idut. p. 135, 8. ' s'h'd, p. 300, 3.

* sHd, p. 341, 19. • i^ut. p. 353, 15.
" Md. p. 363, 15.
" See the passages referred to by van Pesch (p. 70). The most glaring case is a passage

(p. 31, 19) where he quotes Plotinus, using the expression *' Plotinus says " Comparison

with Plotinus, Ennead, I. 3. 3i shows that very few words are those of Plotinus himself; the
rest represent Plotinus* views in Proclus' own language.
" Proclus, p. 330, 19 sqq.

r

CH. iv] PROCLUS AND HIS SOURCES 35

in writing his commentary. Three have already been disposed of,
viz. Heron, Porphyry and Pappus, who had all written commentaries
on the Elements^, We go on to

Eudemus, the pupil of Aristotle, who, among other works, wrote a
history of arithmetic, a history of astronomy, and a history of geometry.
The importance of the last mentioned work is attested by the frequent
use made of it by ancient writers. That there was no other history
of geometry written after the time of Eudemus seems to be proved by
the remark of Proclus in the course of his famous summary : " Those
who compiled histories bring the development of this science up to
this point Not much younger than tliese is Euclid*...." The loss of
Eudemus' history is one of the gravest which fate has inflicted upon
us, for it cannot be doubted that Eudemus had before him a number
of the actual works of earlier geometers, which, as before observed,
seem to have vanished completely when they were superseded by the
treatises of Euclid, Archimedes and Apollonius. As it is, we have to
be thankful for the fragments from Eudemus which such writers as
Proclus have preserved to us.

1 agree with van Pesch* that there is no sufficient reason for
doubting that the work of Eudemus was accessible to Proclus at first
hand. For the later writers Simplicius and Eutocius refer to it in
terms such as leave no room for doubt that thty had it before them.
I have already quoted a passage from Simplicius' account of the lunes
of Hippocrates to the effect that Eudemus must be considered the
best authority since he lived nearer the times^ In the same place
Simplicius says', ** I will set out what Eudemus says word for word
(xariL Xi^ip Xeyofi^va), adding only a little explanation in the shape of
reference to Euclid's Elements owing to the memorandum-like style of
Eudemus {h^a top vwofipnfiaTMOP rpoirop rov EvSif/xou) who sets out
his explanations in the abbreviated form usual with ancient writers.
Now in the second book of the history of geometry he writes as
follows'." It is not possible to suppose that Simplicius would have
written in this way about the style of Eudemus if he had merely been
copying certain passages second-hand out of some other author and
had not the original work itself to refer to. In like manner, Eutocius
speaks of the paralogisms handed down in connexion with the
attempts of Hippocrates and Antiphon to square the circle', "with
which I imagine that those are accurately acquainted who have
examined {iirecKefifiipov^) the geometrical history of Eudemus and
know the Ceria Aristotelica." How could the contemporaries of Euto-
cius have examified the work of Eudemus unless it was still extant in
his time ?

The passages in which Proclus quotes Eudemus by name as his
authority are as follows :

(i) On I. 26 he says that Eudemus in his history of geometry

^ See pp. 10 to 17 above.

• Proclus, p. 68, 4 — 7. ' De Proclifontibus, pp. 73 — 75.

^ See above» p. 19. ^ Simplicius, loc. cit., ed. Diels, p. 60, 17.

• Archimedes, ed. Heiberg, vol. in. p. 164.

36 INTRODUCTION [ch, iv

referred this theorem to Thales, inasmuch as it was necessary to
Thales method of ascertaining the distance of ships from the shore',

(2) Eudemus attributed to Thales the discovery of Eucl L 15',
and

(3) to Oenopides the problem of 1. 23".

(4) Eudemus referred the discovery of the theorem in L 32 to the
Pythagoreans, and gave their proof of it^ which Proclus reproduces*.

(j) On I, 44 Proclus tells us* that Eudemus says that *' these
things are ancient, being discoveries of the Pythagorean muse, the
application (wapa^oXi^) of areas, their exceeding {yirfpffoXf)) and
their falling short (eXXci^K)*" The next words about the appro-
priation of these terms (parabola, hyperbola and ellipse) by later
writers (i.e, Apollonius) to denote the conic sections are of course not
due to Eudemus.

Coming now to notes where Eudemus is not named by Proclus,
we may fairly conjecture, with van Pesch, that Eudemus was really
the authority for the statements (i) that Thales first proved that a
circle is bisected by its diameter" (though the proof by nduciio ad
absurdum which follows in Proclus cannot be attributed to Thales^),
(2) that '* Plato made over to Leodamas the analytical method, by
means of which it is recorded (/txTopi^ra*) that the latter too made
many discoveries in geometry V* (3) that the theorem of I* 5 was due
to Thales, and that for equal angles he used the more archaic
expression "similar'' angles*, (4) that Oenopides first investigated
the problem of L 12. and that he called the perpendicular the
gfiomonic line {Kara yva^fioi'ay^, (5) that the theorem that only three
sorts of polygons can fill up the space round a point, viz. the
equilateral triangle, the square and the regular hexagon, was
Pythagorean", Eudemus may also be the authority for Proclus'
description of the two methods, referred to Plato and Pythagoras
respectively, of forming right-angled triangles in w^hole numbers".

We cannot attribute to Eudemus the beginning of the note on
I, 47 where Proclus says that ''if we listen to those who like to
recount ancient history, we may find some of them referring this
theorem to Pythagoras and saying that he sacrificed an ox in honour
of his discovery"/' As such a sacrifice was contrary to the Pytha-
gorean tenets, and Eudemus could not have been unaware of this,
the story cannot rest on his authority. Moreover Proclus speaks as
though he were not certain of the correctness of the tradition ; indeed,

* FrocliiS, p. 351, 14^ — 18. * i&iJ^ p. 399, 3.

» iSit/. p* 333, 5. * iAid. p. 379, i— 16.

* iM, p, 419, 15—18. * i^id, p. ij7, 10, It.

^ Cvitor (^^fi. d. Alaih, ij, p. %%%) poinis Qui Uic connexion between the nducii^ ad
ahiurdttm and the analyticiit method said to have been discovered by Piato. Proclus gives
the proof by reductio ^ abiurdum to meet an imagiitary critic who d^tres a maihtmatkal
proof ; possibly Thales may have been satisRei! with the argument in the same aeotence
which mentions Thales, ^Mhe cause of the bisection being the unswerving course of the
straight line through the centre."

* Proclus, p. 31I9 19—33. ' i^' p. 350, 30.

*• Md, p. S83, 7 — 10. " iHd. pp. 304, II — 305, 3.

^ iHd. pp. 438, 7— 439, 9. ^ Hid. p. 4s6, 6—9.

1

\

r

!

CH. IV] PROCLUS AND HIS SOURCES 37

so far as the story of the sacrifice is concerned, the same thing is told
of Thales in connexion with his discovery that the angle in a semi-
circle is a right angle*, and Plutarch is not certain whether the ox
was sacrificed on the discovery of I. 47 or of the problem about
application of areas*. Plutarch's doubt suggests that he knew of no
evidence for the story beyond the vague allusion in the distich of
Apollodorus "Logisticus" (the "calculator") cited by Diogenes
Laertius also*; and Proclus may have had in mind this couplet with
the passages of Plutarch.

We come now to the question of the famous historical summary
given by Proclus*. No one appears to maintain that Eudemus is the
author of even the early part of this summary in the form in which
Proclus gives it It is, as is well known, divided into two distinct
parts, between which comes the remark, "Those who compiled
histories' bring the development of this science up to this point.
Not much younger than these is Euclid, who put together the
Elements, collecting many of the theorems of Eudoxus, perfecting
many others by Theaetetus, and bringing to irrefragable demonstration
the things which had only been somewhat loosely proved by his pre-
decessors." Since Euclid was later than Eudemus, it is impossible that
Eudemus can have written this. Yet the style of the summary after
this point does not show any such change from that of the former
portion as to suggest different authorship. The author of the earlier
portion recurs frequently to the question of the origin of the
elements of geometry in a way in which no one would be likely to
do who was not later than Euclid ; and it must be the same hand
which in the second portion connects Euclid's Elements with the
work of Eudoxus and Theaetetus'.

If then the summary is the work of one author, and that author
not Eudemus, who is it likely to have been ? Tannery answers that
it is Geminus' ; but I think, with van Pesch, that he has failed to
show why it should be Geminus rather than another. And certainly
the extracts which we have from Geminus' work suggest that the sort
of topics which it dealt with was quite different ; they seem rather to
have been general questions of the content of mathematics, and even
Tannery admits that historical details could only have come inci-
dentally into the work'.

Could the author have been Proclus himself.^ Circumstances

1 Diogenes Laertius, i. 94, p. 6, ed. Cobet.

' Plutarch, non posse suaviter vivi secundum Epicurum, 1 1 ; Symp, vili, i.

' Diog. LAert vill. 11, p. 107, ed. Cobet:

'HrUa Iiv$ay6f^ rb ve/xjcXe^r c0pero ypdfAfia^
K^uf 4^ Ih-ti) JcXeirV ^*7* fiovOvairfp.
See on this subject Tannery, La CiomJtrie grecque^ p. 105.

• Proclus, pp. 64 — 70.

• The plural is well explained by Tannery, La GiomHrie grecque, pp. 73, 74. No doubt
the author of the summary tried to supplement Eudemus by means of any other histories
which threw light on the subject. Thus e.g. the allusion (p. 64, 11) to the Nile recalls
Herodotus. Cf. the expression in Proclus, p. 64, 19, vapii tup roXXQv irr^ptfrak,

• Tannery, La Gioniitrie grecque^ p. 75.

' ibid, pp. 66—75. • »W. p. 19.

»

38 INTRODUCTION

which seem to suggest this possibility are (i^ that, as already stated, |

the question of the origin of the Ekmiuts is kept prominent, ,

(2) that there is no mention of Democritus, whom Eudemus would
not be likely to have ignored, while a follower of Plato would be
likely enough to do him the injustice, following the example of Plato
who was an opponent of Democntus, never once mentions him, and
is said to have wished to burn all his writings ^ and (3) the allusion at
the beginning to the " inspired Aristotle ** (o tat^jiovto^ 'ApiaTOT€Xf}<;)\
though this may easily have been Inserted by Proclus tn a quotation
made by him from someone else On the other hand there arc
considerations which suggest that Proclus himself was mt the writer,
(i) The style of the whole passage is not such as to point to him {
as the author. (2) If he wrote it, it is hardly conceivable that he
would have passed over in silence the discovery of the analytical
method, the invention of Plato to which he attached so much
importance*.

There is nothing improbable in the conjecture that Proclus quoted
the summary from a compendium of Eudemus' history made by some
later writer: but as yet the question has not been definitely settled.
All that is certain is that the early part of the summary must have
been made up from scattered notices found in the great work of
Eudemus. > [

Proclus refers to another work of Eudemus besides the history,
viz. a book on T/i£ Angle {fiiffktoi/ Trept y^vias)\ Tannery assumes
that this must have been part of the history, and uses this assumption
to confirm his idea that the history was arranged according to subjects,
not according to chronological order*. The phraseology of Proclus
however unmistakably suggests a separate work; and that the
history was chrmmlogicaily arranged seems to be clearly indicated by
the remark of Simplicius that Eudemus **also counted Hippocrates
among the more ancient writers " (cf toX^ TraXatoripo^^)*,

The passage of Simplicius about the lunes of Hippocrates throws
considerable light on the style of Eudemus' history. Eudemus wrote
in a memorandum -like or summary manner (top v^o^vrifiaTLKov rpGirov
rov l&ifB^fiovy when reproducing what he found in the ancient writers ;
sometimes it is clear that he left out altogether proofs or constructions j
of things by no means easy".

Geminus.

The discussions about the date and birthplace of Geminus form a
whole literature, for an account of which I must refer the reader to the (];•
recent edition by Manitius of Gemini elementa astronamiae (Teubner,
1898)'. It must suffice here to state the general conclusion arrived at
by Manitius^^ Though the name looks'^like a Latin name (Geminus),

^ Diog. Laertins, IX. 40, p. 337, ed. Cobet * Proclus, p. 64, 8.

' Proclus, p. an, 19 sqq. ; the passage is quoted above, p. 36.

^ ibid. p. 135, 8. ' Tannery, La Ghmitrie grtcque^ p. 36.

* Simplicius, ed. Diels, p. 69, 33. ' ibid, p. 60, 39.

* Cf. Simplicius, p. 63, 19 sqq. ; p. 64, 35 saq. ; also Usener's note " de supplendis
Hippocratis quas omisit Eudemus constructionibus *' added to Diels' pre£u:e, pp. xziu — xxvL

* See the appendix to this edition, pp. 337 — 353. ^* pp. 351, 353.

CH. ivj PROCLUS AND HIS SOURCES 39

the consistent appearance of it in Greek with the properispomenon
accent (Fe/i&ov) leaves no room for doubt as to its Greek origin,
especially as it is found in inscriptions with the spelling Fc^ielvo^.
The name may be formed from the stem ycfi like ^Epylvo^ from ipy,
*AX€f*i/o9 from oXef. Cf also the unmistakably Greek names 'IktIpo^,
Koarufo^, Geminus, a Stoic philosopher, born probably in the island
of Rhodes, was the author of a comprehensive work on the classifi-
cation of mathematics, and also wrote, about 73-67 B.C., a not less
comprehensive commentary on the meteorological textbook of his
teacher Posidonius of Rhodes.

It is the former work in which we are specially interested here.
Though Proclus made great use of it, he does not mention its title,
unless we may suppose that, in the passage (p. 177, 24) where, after
quoting from Geminus a classification of lines which never meet, he
says, •* these remarks I have selected from the il>iXo/ea\la of Geminus,"
^iXoxaXia is a title or an alternative title. Pappus however quotes a
work of Geminus *'on the classification of the mathematics" (iv r^
w€pi T^9 r£v fiaOrffjidrmv Taf6a)9)^ while Eutocius quotes from "the
sixth book of the doctrine of the mathematics " {iv r^ iKnp t§9 r&v
IJLaOfifiartov OempUn;)*. Tannery* pointed out that the former title
corresponds well enough to the long extract* which Proclus gives in
his first prologue, and also to the fragments contained in the Anonymi
variae colUctiones published by Hultsch at the end of his edition of
Heron* ; but it does not suit most of the other passages borrowed by
Proclus. The correct title was therefore probably that given by
Eutocius, The Doctrine, or Theory, of the Mathematics ; and Pappus
probably refers to one particular portion of the work, say the first
Book. If the sixth Book treated of conies, as we may conclude from
Eutocius, there must have been more Books to follow, because Proclus
has preserved us details about higher curves, which must have come
later. If again Geminus finished his work and wrote with the same
fulness about the other branches of mathematics as he did about
geometry, there must ha(re been a considerable number of Books
altogether. At all events it seems to have been designed to give
a complete view of the whole science of mathematics, and in fact to
be a sort of encyclopaedia of the subject.

I shall now indicate first the certain, and secondly the probable,
obligations of Proclus to Geminus, in which task I have only to follow
van Pesch, who has embodied the results of Tittel's similar inquiry also*.
I shall only omit the passages as regards which a case for attributing
them to Geminus does not seem to me to have been made out.

First come the following passages which must be attributed to
Geminus, because Proclus mentions his name:

(i) (In the first prologue of Proclus^) on the division of mathe-

• Pappus, ed. Hultsch, p. 1026, 9. • Apollonius, ed. Heiberg, vol. ii. p. 170.
' Tannery, La GionUtrU grecque, pp. i8, 19. * Proclus, pp. 38, i — 41, 8.

• Heron, ed. Hultsch, pp. 246, 16--249, '^•

• Van Pesch, De Procti fontibus, pp. 97—113. The disserUtion of Tittel is entitled De
Gemini Stoici studiis nuUhimaticU ( 1 895).

7 Proclus, pp. 38, I — 43, 8, except the allusion in p. 41, 8 — 10, to Ctesibius and Heron and

40 INTRODUCTION [ch. iv

matical sciences into arithmctiCt geometry, mechanics, astronoiny,
optics, geodesy, canonic (science of musical harmony^ and logistic
(apparently arithmetical problems);

(2) (in the note on the definition of a straight line) on the
classification of lines (including curves) as simple (straight or circular)
and mixed, composite and incomposite, uniform (o^oiofttp*!^) and
non-uniform {avo^oiop€p€U\ lines "about solids" and lines produced
by cutting solids, including conic and spiric sections' ;

(3) (i^ the note on the definition of a plane surface) on similar
distinctions extended to surfaces and solids* ;

(4) (in the note on the definition of parallels) on lines which
do not meet {acvpirrmTOi) but which are not on that account
parallel, e,g a curve and its asymptote, showing that the property of
not nueling does not make lines parallel — a favourite observation of
Geminus — and^ incidentally, on bounded lines or those which mciose a
figure and those which do not* ;

(5) (in the same note) the definition of parallels given by
Posidonius*;

(6) on the distinction between postulates and axioms, the futility
of trying to prove axioms, .as ApoUonius tried to prove Axiom i, and
the equal incorrectness of assuming what really requires proof, "as
Euclid did in the fourth postulate [equality of right angles] and in
the fifth postulate [the parallel- postulate]^" ;

(7) on Postulates i, 2, 3, which Geminus makes depend on the
idea of a straight line being described by the motion of a point* ;

(8) (in the note on Postulate 5) on the inadmissibility in geometry
of an argument which is merely plausible, and the danger in this
particular case owing to the existence of lines which do converge
ad infiniium and yet never meet^ ;

(9) (in the note on L i) on the subject-matter of geometry,
theorems, problems and Stopt^fioi (conditions of possibility) for
problems* ;

(10) (in the note on I. 5) on a generalisation of I. 5 by Geminus
through the substitution for the rectilineal base of ** one uniform line
(curve)/' by means of which he proved that the only "uniform lines"

their pneumatic devices {BavnarowoiXiHi), as regards which Proclus' authority may be Pappus
(yni. p. 1094, 74 — ay) who uses very similar exjpressioiis. Heron, even if not later tnan
ueminus, could hardly have been included in a historical work bv him. Perhaps Geminus
may have referred to Ctesibius only, and Proclus may have inserted "and Heron*' himself.

* Prochis, pp. 103, «i— 107, 10; pp. Ill, I— 113, 3.

* tM, pp. riy, 14—110, IS, where perhaps in the passage pp. 117, 33 — ii8, 93 we may
have Gemmus' own words.

* iMd. pp. 176, 18 — 177, 95; perhaps also p. 175. The note ends with the words
"These things too we have selected from Geminus* ^iXoiraXki for the elucidation of the
matters in question.** Tannery ^p. 17) takes these words coming at the end of the commen-
tary on the definitions as referring to the whole of the portion of the commentary dealing
with the definitions. Van Pesch properly regards them as only applying to the note on
paraiUis, This seems to me clear from the use of the word too (ro^'aOra ir aQ.

* Proclus, p. 176, 5—17.

* ihid, pp. 178— 18s, 4; pp. 183, 14— 184, 10; cf. p. 188, 3— II.

* ibid. p. 185, 6 — 15.

' ibid. p. 193, 5 — 19. * iHd, pp. loo, ii — aoi, 35.

CH. iv] PROCLUS AND HIS SOURCES 41

»

, [ (alike in all their parts) are a straight line, a circle, and a cylindrical
t; helix^

(11) (in the note on I. lo) on the question whether a line is made
up of indivisible parts (dfiepr}), as affecting the problem of bisecting
a given straight line* ;

(12) (in the note on I. 35) on topical, or /<?rKJ-theorems*, where
the illustration of the equal parallelograms described between a
hyperbola and its asymptotes may also be due to Geminus*.

Other passages which may fairly be attributed to Geminus, though
his name is not mentioned, are the following :

(i) in the prologue, where there is the same allusion as in the
passage (8) above to a remark of Aristotle that it is equally absurd to
expect scientific proofs from a rhetorician and to accept mere plausi-
bilities from a geometer* ;

(2) a passage in the prologue about the subject-matter, methods,
and bases of geometry, the latter including axioms and postulates* ;

(3) another on the definition and nature of eletnents^ ;

(4) a remark on the Stoic use of the term axiom for every simple
statement {air6<f>avai^ awXriY ;

(5) another discussion on theorems and problems*, in the middle
of which however there are some sentences by Proclus himself".

(6) another passage, in connexion with Def. 3, on lines including
or not including a figure (with which cf part of the passage (4)
above)" ;

(7) a classification of different sorts of angles according as they
are contained by simple or mixed lines (or curves)";

(8) a similar classification of figures", and of plane figures" ;

(9) Posidonius' definition of a figure^* ;

(10) a classification of triangles into seven kinds" ;

(11) a note distinguishing lines (or curves) producible indefinitely
or not so producible, whether forming a figure or not forming a
figure (like the "single-turn spiral")^';

(12) passages distinguishing different sorts of problems", different
sorts of theorems", and two sorts of converses (complete and partial)*;

(13) the definition of the term "porism" as used in the title of
Euclid's Porisms, as distinct from the other meaning of "corollary "";

(14) a note on the Epicurean objection to I. 20 as being obvious
even to an ass" ;

(15) a passage on the properties of parallels, with allusions to

• Proclus, p. 251, 2— II. • ibid. pp. 277, 25 — 370, 11.
» ibid, pp. 394, 11—395, a and p. 395, 13—21. * ibid, p. 395, 8—12.

• ibid. pp. 33, 21—34. I. • ibid. pp. 57, 9—58, 3.
' *^' PP- 72. 3— 75» 4- * if^' p. 77. 3—6-

• ibid. pp. 77, 7—78, 13, and 79, 3—81, 4. " H>id. pp. 78, 13—79* «•
^» ibid. pp. 102, 31—103, '8. " ibid. pp. 116, 7—117, iC
" ibid, pp. 159, II — 160, 9. " ibid, pp. 161, 37—164, 6.
" ibid. p. 143, 5— II. " ibid. p. 168, 4—13.
*' ibid. p. 187, 19—37. w 1^^. pp. 220, 7 — 333, 14; also p. 330, 6—9.
" ibid, pp. 344, 14—346, 13. « ibid, pp. 353, 5—354, 30.
» ibid. pp. 301, 31—303, 13. « ibid, pp. 333, 4— 3«3. 3-

4f INTRODUCTION [cw. iv

Apollonms' C*?*kj, and the curves invented by Nicomedes, Hippias
and Perseus' ;

(t6) a passage on the parallel-postulate regarded as the converse
of L ij\

Of the authors to whom Proclus was indebted in a less degree the
most important is Apollonius of Perga. Two passages allude to his
Ccni€s\ one to a work on irrationals*, and two to a treatise On /A#
cocklias (apparently the cylindrical helix) by Apollonius*. But more
important for our purpose are six references to Apollonius in connexion
with elementary geometry,

(i) He appears as the author of an attempt to explain the idea
of a line (possessing length but no breadth) by reference to daily
experience, e,g. when we tell someone to measure^ merely, the length
of a road or of a wall*; and doubtless the similar passage showing
how we may in like manner get a notion of a surface (without depth)
is his also*.

(2) He gave a new general definition of an angle*, .

(3) He tried to prove certain axioms', and Proclus gives his I
attempt to prove Axiom i, word for word"- J

Proclus further quotes ; \

(4) Apollonius' solution of the problem in Eucl. L lo^ avoiding
Euclid*s use of L 9",

(5) his solution of the problem in l. 11, differing only slightly
from Euclid's*', and

(6) his solution of the problem in l. 23^.
Heiberg" conjectures that Apollonius departed from Euclid's

method in these propositions because he objected to solving problems
of a more general, by means of problems of a more particular*
character. Proclus however considers all three solutions inferior to
Euclid's ; and his remarks on Apollonius' handling of these ele-
mentary matters generally suggest that he was nettled by criticisms
of Euclid in the work containing the things which he quotes from
Apollonius, just as we conclude that Pappus was offended by the
remarks of Apollonius about Euclid's incomplete treatment of the
*• three- and four-line locus"." If this was the case, Proclus can hardly
have got his information about these things at second-hand; and
there seems to be no reason to doubt that he had the actual work of
Apollonius before him. This work may have been the treatise
mentioned by Marinus in the words ''Apollonius in his general
treatise" (*A7roXXioi/io^ iv tJ icaOoKov irpayfjuireiay^ If the notice
in the Fthrist^'' stating, on the authority of Thabit b. Qurra, that

» Proclus, pp. 355, 10--356, 16. « ibid. p. 364. 9—12; pp. 3^4. ao— 365,4.

» ibid. p. 71, lOjL-i*. 356, 8, 6. « ibid. p. 74, 23, 24.

• ibid. pp. i&j, 5, 6, 14, 15. • ibid. p. 100, 5—19.
' ibid. p. 114, ao— 15. • ibid. p. 123, 15—19 (cf. p. 114, 17, p. 115, 17).

• ibid. p. 183. 13, 14. w ibid, pp. 194. as— '95f 5«
" ibid. pp. 179, 16—180, 4. '• ibid, p. 281, 8—19.
" ibid. pp. 335, 16—336, 5. " PkihUgus, vol. XLliI. p. 489.
^ See above, pp. a, 3. >* Marinus in EucHdis Daia^ ed. Menge, p. S34, 16.
» Fikrist^ tr. Sutcr, p. 19.

i

CH. IV] PROCLUS AND HIS SOURCES 43

Apollonius wrote a tract on the parallel-postulate be correct, it may
have been included in the same work. We may conclude generally
that, in it, Apollonius tried to remodel the beginnings of geometry,
reducing the number of axioms, appealing, in his definitions of lines,
surfaces etc., more to experience than to abstract reason, and
substituting for certain proofs others of a more general character.

The probabilities are that, in quoting from the tract of Ptolemy in
which he tried to prove the parallel-postulate, Proclus had the actual
work before him. For, after an allusion to it as **a certain book**'
he gives two long extracts*, and at the beginning of the second
indicates the title of the tract, '* in the (book) about the meeting of
straight lines produced from (angles) less than two right angles," as
he has very rarely done in other cases.

Certain things from Posidonius are evidently quoted at second-
hand, the authority being Geminus (e.g. the definitions of figtire and
parallels) ; but besides these we have quotations from a separate work
which he wrote to controvert Zeno of Sidon, an Epicurean who had
sought to destroy the whole of geometry*. We are told that Zeno
had argued that, even if we admit the fundamental principles (apxai)
of geometry, the deductions from them cannot be proved without the
admission of something else as well, which has not been included in
the said principles*. On I. i Proclus gives at some length the argu-
ments of Zeno and the reply of Posidonius as regards this proposition*.
In this case Zeno's "something else" which he considers to be
assumed is the fact that two straight lines cannot have a common
segment, and then, as regards the "proof" of it by means of the
bisection of a circle by its diameter, he objects that it has been
assumed that two circumferences (arcs) of circles cannot have a
common part Lastly, he makes up, for the purpose of attacking it,
another supposed " proof" of the fact that two straight lines cannot
have a common part. Proclus appears, more than once, to be quoting
the actual words of Zeno and Posidonius ; in particular, two expres-
sions used by Posidonius about "the acrid Epicurean" {jov Bpifiifv
*EinKovp€iopy and his "misrepresentations" (Uoa-eiBwpti^ <fnfai rov
Zi^vtDva avKo<f>avTelvy, It is not necessary to suppose that Proclus
had the original work of Zeno before him, because Zeno's arguments
may easily have been got from Posidonius' reply ; but he would
appear to have quoted direct from the latter at all events.

The work of Carpus mechanicus (a treatise on astronomy) quoted
from by Proclus* must have been accessible to him at first-hand,
because a portion of the extract from it about the relation of theorems
and problems* is reproduced word for word. Moreover, if he were not
using the book itself, Proclus would hardly be in a position to question
whether the introduction of the subject of theorems and problems

• Proclus, p. 191, 13. * ibid, pp. 361, 14— 3^3» ^8» PP- l^hy 7— 3^7» «7-

• ibid, p. 100, 1 — 3. * ibid, pp. 199, 11 — 100, i.

• ibid, pp. 314, 18 — 315, 13; pp. 116, 10 — 118, II.

• ibid, p. 216, 21. ' ibid, p. 218, i.

• ibid, pp. 341, 19 — 343, II. * ibid, pp. 143, 33—343, 11.

L

44 INTRODUCTION [cH. iv

was opportune in the place where it was found (et fiiv mari tcaipiv f}

It is of course evident that Proclus had before him the original
works of PlatOj Aristotle. Archimedes and Plotinus, as well as the
Xvfifittcrd of Porphyry and the works of his master Syrianus (o i5^cTf/>o?
ica0iiyffimpy, from whom he quotes in his note on the definition of an
angle. Tannery also points out that he must have had before him a
group of works representing the Pythagorean tradition on its mystic,
as distinct from its mathematical, side, from Phitolaus downwards, and
comprising the more or less apocryphal Upoi; \oyos of Pythagoras, the
Oracles (Xojta), and Orphic verses".

Besides quotations from writers whom wc can identify with more
or less certainty, there are many other passages which are doubtless
quoted from other commentators whose names we do not know. A
list of such passages is given by van Pesch*, and there is no need to
cite them here.

Van Pesch also gives at the end of his work ' a convenient list of
the books which, as the result of his investigation, he deems to have
been accessible to and directly used by Proclus. The list is worth
giving here, on the same ground of convenience. It is as follows: ^

Eudemus : history of giomttry.

Gem in us : the theory of the mathematical sciejtces,
r Heron : commmlary on the Elements of Euclid.

Porphyry: „ „ „ .

Pappus: „ „ „ j

Apollonius of Perga : a work relating to elementary geometty*

Ptolemy: on the parallel-postulate.

Posidonius : a book controverting Zeno of Sidon.

Carpus ; astronomy,

Syrianus : a discussion on the angle.

Pythagorean philosophical tradition,

Plato s works »

Aristotle's works.

Archimedes' works.

Plotinus: Enneades,
Lastly we come to the question what passages, if any, in the
commentary of Proclus represent his own contributions to the subject
As we have seen, the onus probandi must be held to rest upon him
who shall maintain that a particular note is original on the part of
Proclus. Hence it is not enough that it should be impossible to point
to another writer as the probable source of a note ; we must have a
positive reason for attributing it to Proclus. The criterion must there-
fore be found either (i) in the general terms in which Proclus points
out the deficiencies in previous commentaries and indicates the
respects in which his own will differ from them, or (2) in specific
expressions used by him in introducing particular notes which may

^ Proclus, p. 341, 21, 33. ' ibid, p. 133, 19.

' Tannery, La Giomitrii gncque^ pp. 35, 36.

* Van Pesch, Di Proclifontibus^ p. 139. • ibid. p. 155.

u

CH. iv] PROCLUS AND HIS SOURCES 45

indicate that he is giving his own views. Besides indicating that he
paid more attention than his predecessors to questions requiring
deeper study (ro irpayfunei^he^) and " pursued clear distinctions "
(to evBiaiperop fji^rahiw/covra^y — by which he appears to imply that
his predecessors had confused the different departments of their
commentaries, viz. lemmas, cases, and objections {iparaaci^y — Proclus
complains that the earlier commentators had failed to indicate the
ultimate grounds or causes of propositions*. Although it is from
Geminus that he borrowed a passage maintaining that it is one of the
proper functions of geometry to inquire into causes (r^v airiav icaX
TO tik riy, yet it is not likely that Geminus dealt with Euclid's
propositions one by one ; and consequently, when we find Proclus, on
\\ I. 8, 16, 17, 18, 32, and 47', endeavouring to explain causes, we have
good reiEison to suppose that the explanations are his own.

Again, his remarks on certain things which he quotes from Pappus
can scarcely be due to anyone else, since Pappus is the latest of the
commentators whose works he appears to have used. Under this
head come

(i) his objections to certain new axioms introduced by Pappus*,

(2) his conjecture as to how Pappus came to think of his alterna-
tive proof of I. 5^

(3) an addition to Pappus' remarks about the curvilineal angle
which is equal to a right angle without being one*.

The defence of Geminus against Carpus, who combated his view
of theorems and problems, is also probably due to Proclus*, as well as
an observation on I. 38 to the effect that I. 35 — 38 are really compre-
hended in VI. I as particular cases^*.

Lastly, we can have no hesitation in attributing to Proclus himself
(i) the criticism of Ptolemy's attempt to prove the parallel-postulate >S
and (2) the other attempted proof given in the same note^ (on I. 29)
and assuming as an axiom that '' if from one point two straight lines
forming an angle be produced ad infinitum the distance between them
when so produced ad infinitum exceeds any finite magnitude (i.e.
length)," an assumption which purports to be the equivalent of a
statement in Aristotle". It is introduced by words in which the
writer appears to claim originality for his proof: "To him who
desires to see this proved {KaTaa'tc€va^6fi€vop) let it be said by us
(Keyiado) irap ^fjLoSv)" etc." Moreover, Philoponus, in a note on
Aristotle's Anal, post, I. 10, says that **the geometer (Euclid) assumes
this as an axiom, but it wants a great deal of proof, insomuch that
both Ptolemy and Proclus wrote a whole book upon it"."

> Proclus, p. 84, 13, p. 432, 14,
• ibid, - -- --

CHAPTER V.

THE TEXT",

It is well known that the title of Simson's edition of Euclid (first
brought out in Latin and English in 1756) claims that, in it, "the
errors by which Theon, or others, have long ago vitiated these books
are corrected, and some of Euclid's demonstrations arc restored " ; and
readers of Simson's notes are familiar with the phrase^ used, where
anything in the text does not seem to htm satisfactory, to the effect
that the demonstration has been spoiled, or things have been interpo-
lated or omitted, by Theon "or some other unskilful editor/' Now
most of the MSS. of the Greek text prove by their titles that they
proceed from the recension of the Elements by Theon ; they purport
to be either " from the edition of Theon " [ix t^9 ^iti^vo^ €xSo<r€<a?) or
** from the lectures of Theon *' (agro avvovaimu rov %itapo^). This was
Theon of Alexandria (4th c. A.D,) who also wrote a commentary on
Ptolemy, in which there occurs a passage of the greatest importance
in this connexion*: "But that sectors in equal circles are to one
another as the angles on which they stand Ans been proved by me in
my edition of the Elements at tfu end of the sixth book!' Thus Theon
himself says that he edited the Elements and also that the second part
of VI. 33, found in nearly all the MSS,, is his addition.

This passage is the key to the whole question of Theon*s changes
in the text of Euclid ; for, when Peyrard found in the Vatican the
MS. 190 which contained neither the words from the titles of the other
MSS. quoted above nor the interpolated second part of VI. 33, he was
justified in concluding, as he did, that in the Vatican MS, we have an
edition more ancient than Theon 's. It is also clear that the copyist
of P,or rather of its archetype, had before him the two recensions and
systematically gave the preference to the earlier one ; for at XIII. 6 in
P the first hand has added a note in the margin 1 ** This theorem is
not given in most copies of the new editimt, but is found in those of
the old," Thus we are more fortunate than Simson, since our
judgment of Theon*s recension can be formed on the basis, not of
mere conjecture, but of the documentary evidence afforded by a
comparison of the Vatican MS. just mentioned with what we may
conveniently call, after Heiberg, the Theonine MSS*

* The materkl for the whole af this chapter is Uken firom Heiberg's cdtticm of the
EUmenii^ introduction to vol. V., And itotn tbe same schoUr's Uiierargtsatuhiiuk^ SittdUn
iiber Euklidy p. 174 sqq. ^d Fi^ralipomtna fv Euklidtn Htrma^ XXXV I It.^ 1905.

* I. p. aoj ed. Hainia = p. 50 etL }^iSA\.

CH. v] THE TEXT 47

The MSS. used for Heibei^s edition of the Elements are the
following :

(i) P = Vatican MS. numbered 190, 4to, in two volumes (doubt-
less one originally) ; loth c.

This is the MS. which Peyrard was able to use ; it was sent from
Rome to Paris for his use and bears the stamp of the Paris Imperial
Library on the last page. It is well and carefully written. There are
corrections some of which are by the original hand, but generally in
paler ink, others, still pretty old, by several different hands, or by one
hand with different ink in different places (P m. 2), and others again
by the latest hand (P m. rec). It contains, first, the Elements I. — XIII.
with scholia, then Marinus' commentary on the Data (without the
name of the author), followed by the Data itself and scholia, then the
Elements XIV., XV. (so called), and lastly three books and a part of a
fourth of a commentary by Theon €t9 toi)? irpox^ipov^ icavova^ OtoXc-
liaiov.

The other MSS. are " Theonine."

(2) F = MS. XXVIII, 3, in the Laurentian Library at Florence, 4to ;

loth c.
This MS. is written in a beautiful and scholarly hand and contains
the Elements I. — XV., the Optics and the Phaenomena, but is not well
preserved. Not only is the original writing renewed in many places,
where it had become faint, by a later hand of the i6th c, but the same
hand has filled certain smaller lacunae by gumming on to torn
pages new pieces of parchment, and has replaced bodily certain
portions of the MS., which had doubtless become iil^ible, by fresh
leaves. The larger gaps so made good extend from Eucl. vii. 12 to
IX. 1 5, and from XII. 3 to the end ; so that, besides the conclusion of the
Elements, the Optics and Phaenomena are also in the later hand, and we
cannot even tell what in addition to the Elements I. — XIII. the original
MS. contained. Heiberg denotes the later hand by ^ and observes
that, while in restoring words which had become faint and filling up
minor lacunae the writer used no other MS., yet in the two larger
restorations he used the Laurentian MS. xxviil, 6, belonging to the
13th — 14th c. The latter MS. (which Heiberg denotes by f) was
copied from the Viennese MS. (V) to be described below.

(3) B = Bodleian MS., D'Orville X. i inf. 2, 30, 4to ; A.D. 888.
This MS. contains the Elements I. — xv. with many scholia. Leaves

15 — 118 contain L 14 (from about the middle of the proposition) to
the end of Book VI., and leaves 123 — 387 (wrongly numbered 397)
Books VII. — XV. in one and the same elegant hand (9th c). The
leaves preceding leaf 1 5 seem to have been lost at some time, leaves
' 6 to 14 (containing Elem, I. to the place in I. 14 above referred to)
being carelessly written by a later hand on thick and common parch-
ment (13th c). On leaves 2 to 4 and 122 are certain notes in the
hand of Arethas, who also wrote a two-line epigram on leaf 5, the
greater part of the scholia in uncial letters, a few notes and corrections,
and two sentences on the last leaf, the first of which states that the
MS. was written by one Stephen clericus in the year of the world 6397

{

48 INTRODUCTION [cm. v

(^ 888 A.D,)j while the second records Arethas* own acqiiisition of it
Arethas lived from, say, 865 to 939 A.D. He was Archbishop of
Caesarea and wrote a commentary on the Apocalypse, The portions
of his library which survive are of the greatest interest to palaeography
on account of his exact notes of dates, names of copyists, prices of
parchment etc. It is to him also that we owe the famous Plato MS.
from Pat m OS (Cod. Clarkianus) which was written for him in November
895 »,

(4) V = Viennese MS. Philos, Gn No. 103 ; probably 12th c
This MS* contains 292 leaves. Eucl Ekments h — XV. occupying
leaves 1 to 254, after which come the Optics (to leaf 271), the
PAacn(^fn€fta {mxyWlTLX^d at the end) from leaf 272 to leaf 2S2, and lastly
scholia^ on leaves 283 to 292^ also imperfect at the end. The different
materia] used for different parts and the varieties of handwriting make
it necessary for Heiberg to discuss this MS, at some length*. The
handwriting on leaves 1 to 183 (Book h to the middle of X. 105) and
on leaves 203 to 234 (from XL 31, towards the end of the proposition,
to XIII. 7, a few lines down) is the same ; between leaves 184 and 202
there arc two varieties of handwriting, that of leaves 184 to '189 and
that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in
the same handwriting, changes first gradually into that of leaves 184
to 1 89 and then (verso) into a third more rapid cursive writing which
is the same as that of the greater part of the scholia, and also as that
of leaves 243 and 282, althoughj as these leaves are of different
material, the look of the writing and of the ink seems altered
There are corrections both by the first and a second hand, and scholia
by many hands. On the whole, in spite of the apparent diversity of
handwriting in the MS., it is probable that the whole of it was written
at about the same time, and it may (allowing for changes of material,
ink etc) even have been written by the same man. It is at least
certain that, when the Laurentian M.s* xxviii^O was copied from it, the
whole MS. was in the condition in which it is now, except as regards
the later scholia and leaves 283 to 292 which are not in the laurentian
MS.j that MS. coming to an end where the F/taetwrnena breaks off
abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c.
But it was apparently in two volumes originally, the first-con-
sisting of leaves i to 183 ; and it is certain that it was not all copied
at the same time or from one and the same original. For leaves
1 84 to 202 were evidently copied from two MSS. different both from
one another and from that from which the rest was copied. Leaves
1 84 to the middle of leaf 1 89 (recto) must have been copied from a
MS, similar to P^ as is proved by similarity of readings, though not
from P itself The rest, up to leaf 202, were copied from the Bolc^na
MS, (b) to be mentioned below. It seems clear that the content of
leaves 184 to 202 was supplied from other MSS. because there was a
lacuna in the original from which the rest of V was copied ]

]

<

\

^ See Pauly-WUsowa, Rial-EncydopadU der class. AlUrtumtunssensckaftiHt voL IL, 18961
eibeig, vol v. pp. zxix— xxxiii.

p. 67^^

r'

cav] THE TEXT 49

Heiberg sums up his conclusions thus. The copyist of V first
copied leaves i to 183 from an original in which two quaterniones
were missing (covering from the middle of Eucl. x. 105 to near the
end of XL 31). Noticing the lacuna he put aside one quatemio of the
parchment used up to that point Then he copied onwards from
the end of the lacuna in the original to the end of the Phamomena,
After this he looked about him for another MS. from which to fill up
the lacuna ; finding one, he copied from it as far as the middle of leaf
189 (recto). Then, noticing that the MS. from which he was copying
was of a different class, he had recourse to yet another MS. from which
he copied up to leaf 202. At the sanre time, finding that the lacuna
was longer than he had reckoned for, he had to use twelve more
leaves of a different parchment in addition to the quatemio which he
had put aside. The whole MS. at first formed two volumes (the first
containing leaves i to 183 and the second leaves 184 to 282); then,
after the last leaf had perished, the two volumes were made into one
to which two more quaterniones were also added. A few leaves of the
latter of these two have since perished.

(5) b s MS. numbered 18 — 19 in tlie Communal Library at

Bologna, in two volumes, 4to; nth c.
This MS. has scholia in the margin written both by the first hand
and by two or three later hands ; some are written by the latest hand,
Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas,
14th c) who owned the MS. at one time. It contains {a) in 14 quater-
niones the definitions and the enunciations (without proofs) of the
Elements I. — XIIL and of the Data^ {b) in the remainder of the
volumes the Proem to Geometry (published among the Variae
Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14)
followed by the Elements I.— XIII. (part of XIII. 18 to the end being
missing), and then by part of the Data (from the last three words of
the enunciation of Prop. 38 to the end of the penultimate clause in
Prop. 87, ed. Menge). From XI. 36 inclusive to the end of xil. this
MS. appears to represent an entirely different recension. Heiberg is
compelled to give this portion of b separately in an appendix. He
conjectures that it is due to a Byzantine mathematician who thought
j- Euclid's proofs too long and tiresome and consequently contented
11 himself with indicating &e course followed^ At the same time this
Byzantine must have had an excellent MS. before him, probably of the
. ante-Theonine variety of which the Vatican MS. 190 (P) is the sole
representative.

(6) p = Paris MS. 2466, 4to; 12th c.

This manuscript is written in two hands, the finer hand occupying
leaves i to 53 (recto), and a more careless hand leaves 53 (verso) to
64, which are of the same parchment as the earlier leaves, and leaves
I 65 to 239, which are of a thinner and rougher parchment showing
traces of writing of the 8th — 9th c. (a Greek version of the Old
Testament). The MS. contains the Elements I. — xill. and some scholia
after Books xi., XII. and Xlll.

> Ztiischrifi fiir Math, u. Physik^ XXIX., hist.-litt. Abtheilung, p. 13.
H. E. 4

I-

50 INTRODUCTION [cK, v

(7) q = Pans MS. 2344, folio ; 12th c
It is written by one hand but includes scholia by many hands.

On leaves l to 16 (recto) are scholia with the same title as that found
by Wachsmuth in a Vatican MS. and relied upon by him to prove that
Proclus continued his commentaries beyond Book L' Leaves 17 to
357 contain the Elements L — XIIL (except that there is a lacuna from
the middle of VIIL 25 to the Sxffetrt*^ of IX, 14); before Books VIL and
X, there are some leaves filled with scholia only* and leaves 358 to 366
contain nothing but scholia,

(8) Heiberg also used a palimpsest in the British Museum (Add,
1 72 11). Five pages are of the 7th — 8th c, and are contained (leaves
49 — S3) in the second volume of the Syrian ms. Brit. Mus. 687 of the
9th c, ; half of leaf 50 has perished » The leaves contain various frag-
ments from Book X. enumerated by Heiberg, Vol. III., p, v, and nearly
the whole of X II I, 14,

Since his edition of the Elements was published, Heiberg has
collected further material bearing on the history of the text*. Besides
giving the results of further or new examination of MSS,. he has
collected the fresh evidence contained in an-Nairizl's comment a r>\
and particularly in the quotations from Heron *3 commentary given in
it (often word for word^ which enable us in several cases to trace I
differences between our text and the text as Heron had it, and to
identify some interpolations which actually found their way into the
text from Heron's commentary itself; and lastly he has dealt with
some valuable fragments of ancient papyri which have recently come
to light, and which are especially important in that the evidence drawn
from them necessitates some modification in the views expressed in
the preface to VoL V, as to the nature of the changes made in Theon's
recension, and in the principles laid down for differentiating between
Theon's recension and the original text, on the basis of a comparison
between P and the Theonine MSS. alone*

The fragments of ancient papyri referred to are the following,

I* Pafyrus Herculanensis No. 1061*

This fragment quotes Def. i j of Book l. in Greek, and omits the
words ^ ^aX^trat 7r€pi<f>ip€ta^ "which is called the circumference/'
found in all our MSS., and the further addition ^/>ot rtju roi) /WKXav
TTipitf^tp^tav also found in practically all the MSS. Thus Heibei^'s
assumption that both expressions are interpolations is now confirmed
by this oldest of all sources,

2. T^ Oxyrhymhus Papyri 1. p. 58, No. XXIX* of the 3rd or 4th c.

This fragment contains the enunciation of Eucl IL 5 (with figure,
apparently without letters, immediately following, and not, as usual in
our MSS., at the end of the proof) and before it the part of a word""
vepiexofie belonging to II. 4 (with room for -yf» opOo^vUp* iirep l^i

^ [tit rjk T9O EAxXcldov rrocxcM wpdXa^ifiaifi/uwa 4k rQif np6ffXov aw^fdimif mi car* fri-
n^^. Cf p. 3a, note 8, above.

' Heibeig, Faratipomtna mm EukHd in Hermes^ xxxviiu, 15^3, pp. 46 — 74, 161 — aoi,

' Described by Heibeig in Otfersigt aver dit kmgi. damJki VuUmkabemis Siiskttbs
Forhandtingpr^ 1900, p. 161.

. .

("

CH.V] THE TEXT 51

Set^ai and a stroke to mark the end), showing that the fragment had
not the Porism which appears in all the Theonine MSS. and (in a later
hand) in P, and thereby confirming Heiberg's assumption that the
Porism was due to Theon.

3. A fragment in Fayum towns and ttieir papyri^ p. 96, No. IX. of
2nd or 3rd c

This contains I. 39 and I. 41 following one another and almost
complete, showing that I. 40 was wanting, whereas it is found in all
the MSS. and is recognised by Proclus. Moreover the text of the
beginning of I. 39 is better than ours, since it has no double hiopur/AS^
but omits the first (" I say that they are also in the same parallels ")
and has " and** instead of **/or let ^Z> be joined " in the next sentence.
It is clear that I. 40 was interpolated by someone who thought there
ought to be a proposition following I. 39 and related to it as I. 38 is
related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses I. 40,
and therefore was not likely to include it The same interpolator
failed to realise that the words "let AD he joined" were part of the
i/cOeai^ or setting-out, and took them for the tcaraa-tcewj or " construc-
tion " which generally follows the hiopurfio^ or " particular statement "
of the conclusion to be proved, and consequently thought it necessary
to insert a Siopurfio^ before the words.

The conclusions drawn by Heiberg from a consideration of
particular readings in this papyrus along with those of our MSS. will
be referred to below.

We now come to the principles which Heiberg followed, when
preparing his edition, in differentiating the original text from the
Theonine recension by means of a comparison of the readings of P
and of the Theonine MSS. The rules which he gives are subject to a
certain number of exceptions (mostly in cases where one Ms. or the
other shows readings due to copyists' errors), but in general they may
be relied upon to give conclusive results'.

The possible alternatives which the comparison of P with the
Theonine MSS. may give in particular passages are as follows :

I. There may be agreement in three different degrees.

(i) P and all the Theonine MSS. may agree.

In this case the reading common to all, even if it is corrupt or
interpolated, is more ancient than Theon, i.e. than the 4th c.

(2) P may agree with some (only) of the Theonine MSS.

In this case Heiberg considered that the latter give the true
reading of Theon's recension, and the other Theonine MSS. have
departed from it

(3) P and one only of the Theonine MSS. may agree.

In this case too Heiberg assumed that the one Theonine MS. which
agrees with P gives the true Theonine reading, and that this rule even
supplies a sort of measure of the quality and faithfulness of the
Theonine MSS. Now none of them agrees alone with P in preserving
the true reading so often as F. Hence F must be held to have pre-
served Theon 's recension more faithfully than the other Theonine MSS.;
and it would follow that in those portions where F fails us P must

4—2

L

$2 INTRODUCTION [ch.

cany rather more weight even though it may differ from the Theonine
MSS. BVpq. (Heiberg gives many examples en proof of this, as of his
main rules generally, for which reference must be made to his Prole-
gomena in Vol V.) The specially close relation of F and P is also
illustrated by passages in which they have the same errors \ the
explanation of these common errors (where not due to accident) is
found by Heiberg in the supposition that they existed, but were not
noticed by Theon, in the original copy in which he made his changes.

Although however F is by far the best of the Theonine MSS., there
are a considerable number of passages where one of the others (B, V,
p or q) alone with P gives the genuine reading of Theon's recension.

As the result of the discovery of the papyrus fragment containing
I. 39, 41, the principles above enunciated under (2) and (3) are found
by Heiberg to require some qualification. For there is in some cases
a remarkable agreement between the papyrus and the Theonine MSS.
(some or all) as against R This shows that Theon took more trouble
to follow older MSS,, and made fewer arbitrary changes of his own,
than has hitherto been supposed. Next, when the papyrus agrees
with some of the Theonine MSS. against P, it must now be held that
these MSS. (and not, as formerly supposed, those which agree with P)
give the true reading of Theon. If it were otherwise, the agreement
between the papyrus and the Theonine MSS. would be accidental: but
it happens too often for this. It is clear also that there must have
been contamination between the two recensions ; otherwise, whence
could the Theonine MSS, which agree with P and not with the papyrus
have got their readings? The influence of the P class on the Theonine
F is especially marked.

11. There may be disagreement between P and all the Theonine
MSS.

The following possibilities arise.

(1) The Theonine MSS. differ also among themselves*

In this case Heiberg considered that P nearly always has the true
reading, and the Theonine MSS, have suffered interpolation in different
ways after Theon's time.

(2) The Theonine Mss. all combine against P.

In this case the explanation was assumed by Heiberg to be one or
other of the following.

(a) The common reading is due to an error which cannot be
imputed to Theon (though it may have escaped him when putting
together the archetype of his edition); such error may either have
arisen accidentally in all alike, or (more frequently) may be
referred to a common archetype of all the MSS.

(/3) There may be an accidental error in P ; e.g. something
has dropped out of P in a good many places, generally through

(<y) There may be words interpolated in P.
(0) Lastly, we may have in the Theonine MSS. a change made
by Theon himself,
(The discovery of the ancient papyrus showing readings agreeing

i

CH. v] THE TEXT 53

with some, or with all, of the Theonine MSS. against P now makes it
necessary to be very cautious in applying these criteria.)

It is of course the last class (8) of changes which we have to
investigate in order to get a proper idea of Theon's recension.

Heiberg first observes, as regards these, that we shall find that
Theon, in editing the Elements, altered hardly anything without some
reason, often inadequate according to our ideas, but still some reason
which seemed to him sufficient Hence, in cases of very slight differ-
ences where both the Theonine MSS. and P have readings good and
probable in themselves, Heiberg is not prepared to put the differences
down to Theon. In those passages where we cannot see the least
reason why Theon, if he had the reading of P before him, should have
altered it, Heiberg would not at once assume the superiority of P
unless there was such a consistency in the differences as would indicate
that they were due not to accident but to design. In the absence of
such indications, he thinks that the ordinary principles of criticism
should be followed and that proper weight should be attached to the
antiquity of the sources. And it cannot be denied that the sources of
the Theonine version are the more ancient. For not only is the
British Museum palimpsest (L), which is intimately connected with
the rest of our MSS., at least two centuries older than P, but the other
Theonine MSS. are so nearly allied that they must be held to have
had a common archetype intermediate between them and the actual
edition of Theon ; and, since they themselves are as old as, or older
than P, their archetype must have been much older. Heiberg gives
(pp. xlvi, xlvii) a list of passages where, for this reason, he has
followed the Theonine MSS. in preference to P.

It has been mentioned above that the copyist of P or rather of its
archetype wished to give an ancient recension. Therefore (apart from
clerical errors and interpolations) the first hand in P may be relied
upon as giving a genuine reading even where a correction by the first
hand has been made at the same time. But in many places the first
hand has made corrections afterwards ; on these occasions he must
have used new sources, e.g. when inserting the scholia to the first
Book which P alone has, and in a number of passages he has made
additions from Theonine MSS.

We cannot make out any " family tree " for the different Theonine
MSS. Although they all proceeded from a common archetype later
than the edition of Theon itself, they cannot have been copied one
from the other ; for, if they had been, how could it have come about
that in one place or other each of them agrees alone with P in pre-
serving the genuine reading } Moreover the great variety in their
agreements and disagreements indicates that they have all diverged
to about the same extent from their archetype. As we have seen that
P contains corrections from the Theonine family, so they show correc-
tions from P or other MSS. of the same family. Thus V has part of
the lacuna in the MS. from which it was copied filled up from a MS.
similar to P, and has corrections apparently derived from the same ;
the copyist, however, in correcting V, also used another MS. to which

1

54 INTRODUCTION [ch, v

he alludes in the additions to ix, 19 and 30 (and also on X, 23 For);
"in the book of the Ephesian (this) is not found*' Who this Ephesian
of the 1 2th c was, we do not know.

We now come to the alterations made by Theon in his edition of
the Elements. I shall indicate classes into which these alterations
may be divided but without details (except in cases where they affect
the mathematical contmt as distinct from form or language pure and
simple)*.

I. Alterations made by Theon where he founds or thought kt founds
mistakes in the original

1. Real blots in the original which Theon saw and tried to
remove.

{a) Euclid has a porism (corollary) to vi. 19, the enunciation
of which speaks of similar and similarly described figures though the
proposition itself refers only to triangles, and therefore the porism
should have come after VI. 20. Theon substitutes triangle for figtirt
and proves the inore general porism after VL 20.

Qf) In IX. 19 there is a statement which is obviously incorrect.
Theon saw this and altered the proof by reducing four alternatives to
two, with the result that it fails to correspond to the enunciation even
with Theon's substitution of** if*' for '* when " in the enunciation,

{c) Theon omits a porism to IX* 11, although it is necessary for
the proof of the succeeding proposition, apparently because, owing to
an error in the text {Kmrk ^qv corrected by Heiberg into hti th)^ he
could not get out of it the right sense.

(rf) I should also put into this categoiy a case which Heiberg
classifies among those in which Th^:^n merely fancied that he found
mistakes, viz. the porism to V. 7 stating that, if four magnitudes are
proportional, they are proportional inversely. Theon puts this after
V. 4 with a proof, which however has no necessary connexion with
V. 4 but is obvious from the definition of proportion,

{f) I should also put under this head Xl. i, where Euclid s argu-
ment to prove that two straight lines cannot have a common segment
is altered.

2. Passages which seemed to Theon to contain blots, and which
he therefore set himself to correct, though more careful consideration
would have shown that Euclid's words are right or at least may be
excused and offer no difficulty to an intelligent reader. Under this
head come :

(a) an alteration in ill. 24.

\b) a perfectly unnecessary alteration, in vi. 14, of "equiangular
parallelograms" into "parallelograms having one angle equal to one
angle,'* where Theon followed the false analogy of vi. 1 5.

{c) an omission of words in v. 26, owing to his having been mis-
led by a wrong figure.

(rf) an alteration of the order of XI. Deff. 27, 28.

\e) the substitution of " parallelepipedal solid " for " cube *' in XI.

^ Exhaustive detaUs mider aU the difleient heads are given by Heibeig (VoL v.
pp. lii— bov).

CH. v] THE TEXT 55

38, because Theon observed, correctly enough, that it was true of the
parallelepipedal solid in general as well as of the cube, but failed to
give weight to the fact that Euclid must have given the particular
case of the cube for the simple reason that that was all he wanted for
use in xill. 17.

(/) the substitution of the letter ^ for ft ( F for Z in my figure)
because he saw that the perpendicular from K to B^ would fall on ^
itself, so that ^, ft coincide. But, if the substitution is made, it should
ht proved that 4>, ft coincide. Euclid can hardly have failed to notice
the fact, but it may be that he deliberately ignored it as unnecessary
for his purpose, because he did not want to lengthen his proposition
by giving the proof

II. Emendations intended to improve the form or diction of Euclid.
Some of these emendations of Theon affect passages of appreciable

length. Heiberg notes about ten such passages ; the longest is
^ in Eucl. XII. 4 where a whole page of Heiberg's text is affected and
I Theon's version is put in the Appendix. The kind of alteration may
be illustrated by that in ix. 15 where Euclid uses successively the
propositions Vll. 24, 25, quoting the enunciation of the former but not
of the latter ; Theon does exactly the reverse. In a few of the cases
here quoted by Heiberg, Theon shortened the original somewhat.

But, as a rule, the emendations affect only a few words in each
sentence. Sometimes they are considerable enough to alter the con-
formation of the sentence, sometimes they are trifling alterations
"more magistellorum ineptorum" and unworthy of Theon. Generally
speaking, they were prompted by a desire to change anything which
was out of the common in expression or in form, in order to reduce
the language to one and the same standard or norm. Thus Theon
changed the order of words, substituted one word for another where
the latter was used in a sense unusual with Euclid (e.g. iweiiiprcp,
"since," for Sn in the sense of "because"), or one expression for
another in like circumstances (e.g. where, finding ''that which was
enjoined would be done" in a theorem, Vll. 31, and deeming the phrase
more appropriate to a problem, he substituted for it " that which is
sought would be manifest"; probably also and for similar reasons he
made certain variations between the two expressions usual at the end
of propositions oirtp Ihei Sel^ai and Swep eSet iroifjaai, quod erat
demonstrandum and quod erat faciendum). Sometimes his alterations
show carelessness in the use of technical terms, as when he uses
iiTTcaOai (to meet) for i^dtrr^cBai (to touch) although the ancients
carefully distinguished the two words. The desire of keeping to a
standard phraseology also led Theon to omit or add words in a
number of cases, and also, sometimes, to change the lettering of
figures.

But Theon seems, in editing the Elements, to have bestowed the
most attention upon

III. Additions designed to supplement or explain Euclid,
First, he did not hesitate to interpolate whole propositions where

be thought there was room or use for them. We have already

56 INTRODUCTION

mentioned the addition to vi. 33 of the second part relating to sectors^
for which Theon himself takes credit in his commentary on Ptolemy.
Again, he interpolated the proposition commoniy known as VI L 22
{ex aequo in propartione perturbata for numbers, corresponding to V, 23X
and perhaps also vil 20, a particular case of viL 19 as VL 17 is of VL
16. He added a second case to VL 27, a porism to IL 4, a second
porism to III. 16, and a lemma after X. 12; perhaps also the porism
to V. 19 and the first porism to vi, 2a He also inserted alternative
proofs here and there, e.g, in IL 4 (where the alternative differs little
from the original) and in vn. 31 ; perhaps also in X. i, 6, and 9*

Secondly, he sometimes repeats an argument where Euclid had
said "For the same reason/* adds specific references to points,
straight lines etc. in the figures in order to exclude the possibility
of mistake arising from Euclid's reference to them in general terms,
or inserts words to make the meaning of Euclid more plain, e.g.
componendo and alternately^ where Euclid had left them out. Some-
times he thought to increase by his additions the mathematical
precision of Euclid^s language in enunciations or elsewhere, sometimes
to make smoother and clearer things which Euclid had expressed
with unusual brevity and harshness or carelessness, in reliance on the
intelligence of his readers*

Thirdly, he supplied intermediate steps where EucHd^s argument
seemed too rapid and not easy enough to follow. The form of these
additions varies; they are sometimes placed as a definite intermediate
step with "therefore*' or "so that" sometimes they are additions to
the statement of premisses, sometimes phrases introduced by ** since "
"for" and the like, after the inference.

Lastly, there Is a very large class of additions of a word, or one
or two words, for the sake of clearness or consistency. Heiberg
gives a number of examples of the addition of such nouns as
"triangle," "square," ''rectangle,** "magnitude/* "number" "pointy"
"side," "circle," "straight line" "area" and the like, of adjectives
such as " remaining/' " right," " whole/' " proportional/' and of other
parts of speech, even down to words like " is " {i<irl) which is added
600 times. Si;, &pa^ ^ikp, ydp, scat and the like.

IV. Omissions by Tksan.

Heibei^ remarks that, Theon *s object having been, as above
shown, to amplify and explain Euclid, we should not naturally have
expected to find him doing much in the contrary process of com-
pression, and it is only owing to the recurrence of a certain sort of
omissions so frequently (especially in the first Books) as to exclude
the hypothesis of their being all due to chance that we are bound to
credit him with alterations making for greater brevity. We have
seen, it is true, that he made omissions as well as additions for the
purpose of reducing the language to a certain standard form. But
there are also a good number of cases where in the enunciation of
propositions, and in the exposition (the re-statement of them with
reference to the figure), he has left out words because, apparently,
he regarded Euclid's language as being too careful and precise.

CH.V] THE TEXT 57

Again, he is apparently responsible for the frequent omission of the
words iir^p ioei Bel^i (or 'n'oi,fiaa4)t Q.E.D. (or F.), at the end of
propositions. This is often the case at the end of porisms, where,
in omitting the words, Theon seems to have deliberately departed
from Euclid's practice. The MS. P seems to show clearly that, where
Euclid put a porism at the end of a proposition, he omitted the
Q.E.D. at the end of the proposition but inserted it at the end of the
porism, as if he regarded die latter as being actually a part of the
proposition itself. As in the Theonine MSS. the Q.E.D. is generally
omitted, the omission would seem to have been due to Theon.
Sometimes in these cases the Q.E.D. is interpolated at the end of the
proposition.

Heiberg summed up the discussion of Theon's edition by the
remark that Theon evidently took no pains to discover and restore
from MSS. the actual words which Euclid had written, but aimed
much more at removing difficulties that might be felt by learners
in studying the book. His edition is therefore not to be compared
with the editions of the Alexandrine grammarians, but rather with
the work done by Eutocius in editing Apollonius and with an
interpolated recension of some of the works of Archimedes by a
certain Byzantine, Theon occupying a position midway between these
two editors, being superior to the latter in mathematical knowledge
but behind Eutocius in industry (these views now require to be some-
what modified, as above stated). But however little Theon's object
may be approved by those of us who would rather know the
ipsissima verba of Euclid, there is no doubt that his work was
approved by his pupils at Alexandria for whom it was written ; and
his edition was almost exclusively used by later Greeks, with the
result that the more ancient text is only preserved to us in one MS.

As the result of the above investigation, we may feel satisfied
that, where P and the Theonine MSS. agree, they give us (except in a
few accidental instances) Euclid as he was read by the Greeks of
the 4th c. But even at that time the text had been passed from
hand to hand through more than six centuries, so that it is certain
that it had already suffered changes, due partly to the fault of
copyists and partly to the interpolations of mathematicians. Some
errors of copyists escaped Theon and were corrected in some MSS.
by later hands. Others appear in all our MSS. and, as they cannot
have arisen accidentally in all, we must put them down to a common
source more ancient than Theon. A somewhat serious instance is
to be found in III. 8 ; and the use of amriadta for i^irriaOta in the
sense of "touch" may also be mentioned, the proper distinction
between the words having been ignored as it was by Theon also.
But there are a number of imperfections in the ante-Theonine text
which it would be unsafe to put down to the errors of copyists, those
namely where the good MSS. agree and it is not possible to see any
motive that a copyist could have had for altering a correct reading.
In these cases it is possible that the imperfections are due to a
certain degree of carelessness on the part of Euclid himself; for it

S8 INTRODUCTION [ch. v

is not p(^sib1e ** Euclidem ab omni naevo vindicare/' to use the
words of SaccheriSand consequently Simson ts not right in attributing
to Theon and other editors all the things in EucHd to which mathe-
matical objection can be taken. Thiis^ when EucHd speaks of "the
ratio compounded of the sides" for *'the ratio compounded of the
ratios of the sides/* there is no reason for doubting that Euclid himself
is responsible for the more slip-shod expression. Again, in the Books
XI. — XIIL relating to solid geometry there arc blots neither few
nor altc^ether unimportant which can only be attributed to Euclid
himself*; and there is the less reason for hesitation in so attributing
them because solid geometry was then being treated in a thoroughly
systematic manner for the first time. Sometimes the cofulusion
{cvfvrripcLciia) of a proposition does not correspond exactly to the
enunciation, often it is cut short with the w^ords koX ra efij? " and the
rest" (especially from Book X. onwards), and very often in Books VIII*,
IX. it is omitted. Where all the MSS, agree, there is no ground for
hesitating to attribute the abbreviation or omission to Euclid ; though,
of course, where one or more Ms*>. have the longer form, it must be
retained because this is one of the cases where a copyist has a
temptation to abbreviate.

Where the true reading is preserved in one of the Theonine MSS,
alone, Heibcrg attributes the wrong reading to a mistake which arose
before Theon's time, and the right reading of the single MS, to a
successful correction*

We now come to the most important question of the Interpolations
introduced befare Tkeatis time.

I. Alternative proofs or additional cases*

It is not in itself probable that Euclid would have given two
proofs of the same proposition ; and the doubt as to the genuineness
of the alternatives is increased when we consider the character of
some of them and the way in which they are introduced. First of
all, we have those of VL 20 and xn. 17 introduced by '*we shall prove
this otherwise more readily (wpox^ipojef^opy* or that of X, 90 " it is
possible to prove mare slwrtly (<yvifrofiwr€popy Now it is impossible
to suppose that Euclid would have given one proof as that definitely
accepted by him and then added another with the express comment
that the latter has certain advantages over the former. Had he con-
sidered the two proofs and come to this conclusion, he would have
inserted the latter in the received text instead of the former. These
alternative proofs must therefore have been interpolated. The same
argimient applies to alternatives introduced with the words " or even
thus" (fj «cai o{hrfl09)> ''or even otherwise" (tf teal aXXo^). Under this
head come the alternatives for the last portions of IIL 7, 8 ; and
Heibcrg also compares the alternatives for parts of IIL 31 (that the
angle in a semicircle is a right angle) and xill. 18, and the alternative
proof of the lemma after x. 32. The alternatives to x. 105 and 106,

^ EmcUda ab cmm uoivo vindUatus^ Mediolani, T733.

* Cf. espediUy the anomption, without proof or definition, of the criterion for eqtml solid
aa^et, and the incomplete proof of xn. 17.

I CH. V

f aeaii

CH. v] THE TEXT 59

again, are condemned by the place in which they occur, namely after
an alternative proof to X. 115. The above alternatives being all
admitted to be spurious, suspicion must necessarily attach to the few
others which are in themselves unobjectionable. Heibei^ instances
the alternative proofs to III. 9, III. 10, VI. 30, VI. 3 1 and XI. 22, observing
that it is quite comprehensible that any of these might have occurred
to a teacher or editor and seemed to him, rightly or wrongly, to be
better than the corresponding proofs in Euclid. Curiously enough,
Simson adopted the alternatives to HI. 9, 10 in preference to the
genuine proofs. Since Heiberg's preface was written, his suspicion
has been amply confirmed as regards III. 10 by the commentary of
an-NairlzI (ed. Curtze) which shows not only that this alternative is
Heron's, but also that the substantive proposition III. 12 in Euclid
is also Heron's, having been given by him to supplement III. 11
which must originally have been enunciated of circles " touching one
another" simply, i.e. so as to include the case of external as well as
internal contact, though the proof covered the case of internal contact
only. "Euclid, in the iith proposition," says Heron, "supposed two
circles touching one another internally and wrote the proposition on
this case, proving what it was required to prove in it But I will
show haw it is to be proved if the contact be extemalK** This additional
proposition of Heron's is by way of adding another case^ which brings
us to that class of interpolation. It was the practice of Euclid and
the ancients to give only one case (generally the most difficult one)
and to leave the others to be investigated by the reader for himself.
One interpolation of a second case (vi. 27) is due, as we have seen,
to Theon. The two extra cases of XI. 23 were manifestly interpolated
before Theon's time, for the preliminary distinction of three cases,
** (the centre) will either be within the triangle LMN^ or on one of
the sides, or outside. First let it be within," is a spurious addition
(B and V only). Similarly an unnecessary case is interpolated in
III. II.

H. Lemmas.

Heiberg has unhesitatingly placed in his Appendix to Vol. III.
certain lemmas interpolated either by Theon (on X. 13) or later
writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas).
But we are here concerned with the lemmas found in all the MSS.,
which however are, for different reasons, necessarily suspected. We
will deal with the Book X. lemmas last

(i) There is an a priori ground of objection to those lemmas
which come after the propositions to which they relate and prove
properties used in those propositions ; for, if genuine, they would be a
sign of faulty arrangement such as would not be likely in a systematic
work so carefully ordered as the Elements. The lemma to VI. 22 is
one of this class, and there is the further objection to it that in VI. 28
Euclid makes an assumption which would equally require a lemma
though none is found. The lemma after XII. 4 is open to the further
objections that certain altitudes are used but are not drawn in the

^ An-Nairld, ed. Curtze, p. isi.

6o ^ INTRODUCTION [cH. v

figure (which is not in the manner of Euclid), and that a peculiar
expression ** parallele pi pedal solids described on (apaypatf^ofieva aw6)
pristns** betrays a hand other than Euclid's. There is an objection on
the score of language to the lemma after xni. 2. The lemmas on
XL 23, XIII. 13, XIIL 18, besides coming after the propositions to
which they relate, are not very necessary in themselves and, as regards
the lemma to XIIL 13, it is to be noticed that the writer of a gloss
in the proposition could not have had it, and the words "as will
be proved afterwards** in the text are rightly suspected owing to
differences between the MS, readings. The lemma to Xll* 2 also, to
which Simson raised objection, comes after the proposition ; but, if it
is rejected, the words ** as was proved before" used in XIL 5 and i8^
and referring to this lemma, must be struck out

(2) Reasons of substance are fatal to the lemma before X 60,
which is really assumed in X. 44 and therefore should have appeared
there if anywhere, and to the lemma on X. 20, which tries to prove
what is already stated in X. Det 4.

We now come to the remaining lemmas in Book X., eleven in
number, which come before the propositions to which they relate and
remove difficulties in the way of their demonstration. That before
X, 42 introduces a set of propositions with the words *' that the said
irrational straight lines are uniquely divided >.,we will prove after
premising the following lemma," and it is not possible to suppose
that these words are due to an interpolator ; nor are there any
objections to the lemmas before x, 14, 17^ 22, 33, 54, except perhaps
that they are rather easy. The lemma before X. 10 and X io itself
should probably be removed from the Elements ; for X, iO really uses
the following proposition X. il, which is moreover numbered 10 by
the firsthand in P, and the words in X, lo referring to the lemma "for
we learnt (how to do this) '* betray the interpolator Heiberg gives
reason also for rejecting the lemmas before X. 19 and 24 with the
words "in any of the aforesaid ways " (omitted in the Theonine MSS.)
in the enunciations of x. 19, 24 and in the exposition of X. 20. Lastly,
the lemmas before x, 29 may be genuine, though there is an addition
to the second of them which is spurious.

Heiberg includes under this heading of interpolated lemmas two
which purport to be substantive propositions, XL 38 and XIIL 6, These
must be rejected as spurious for reasons which will be found in detail
in my notes on XL 37 and xml 6 respectively. The latter proposition
is only quoted once (in XIii. 17); probably the words quoting it
(with 7/9a/i/Ai^ instead of eiOeia) are themselves interpolated, and
Euclid thought the fact stated a sufficiently obvious inference from
Xlll. I.

III. Porisms (or corollaries).

Most of the porisms in the text are both genuine and necessary ;
but some are shown by differences in the MSS. not to be so, e.g. those
to L 15 (though Proclus has it), III. 31 and VL 20 (For. 2). Sometimes
parts of porisms are interpolated. Such are the last few lines in
tlie porisms to iv. 5, yL 8 ; the latter addition is proved later by

CH. v] THE TEXT 6i

means of vi. 4, 8, so that the writer of these proofs could not have had
the addition to vi. 8 For. before him. Lastly, interpolators have added
a sort of proof to some porisms, as though they were not quite
obvious enough ; but to add a demonstration is inconsistent with the
idea of a porism, which, according to Proclus, is a by-product of a
proposition appearing without our seeking it
IV. Scholia.

Several interpolated scholia betray themselves by their wording,
e.g. those given by Heibei^ in the Appendix to Book X. and contain-
ing the words KoKel^ iKoKeat ("he calls" or "called"); these scholia were
apparently written as marginal notes before Theon's time, and, being
I adopted as such by Theon, found their way into the text in P and
I some of the Theonine MSS. The same thing no doubt accounts for
I the interpolated analyses and syntheses to xiil. i — 5, as to which see
\ my note on xill. i.
f V. Interpolations in Book X.

First comes the proposition " Let it be proposed to us to show that
in square figures the diameter is incommensurable in length with the
side," which, with a scholium after it, ends the tenth Book. The form
of the enunciation is suspicious enough and the proposition, the proof
of which is indicated by Aristotle and perhaps was Pythagorean, is
perfectly unnecessary when X. 9 has preceded. The scholium ends
with remarks about commensurable and incommensurable solids,
which are of course out of place before the Books on solids. The
scholiast on Book x. alludes to this particular scholium as being due
to " Theon and some others." But it is doubtless much more ancient,
and may, as Heiberg conjectures, have been the b^inning of
Apollonius' more advanced treatise on incommensurables. Not only
is everything in Book X. after X. 1 15 interpolated, but Heiberg doubts
the genuineness even of x. 112 — 115, on the ground that x. in
rounds ofT the theory of incommensurables as we want it in the Books
on solid geometry, while x. 112 — 115 are not really connected with
what precedes, nor wanted for the later Books, but seem to form the
starting-point of a new and more elaborate theory of irrationals.

VI. Other minor interpolations are found of the same character as
those above attributed to Theon. First there are two places (XL 35
and XI. 26) where, after "similarly we shall prove " and "for the same
reason," an actual proof is nevertheless given. Clearly the proofs are
interpolated; and there are other similar interpolations. There
are also interpolations of intermediate steps in proofs, unnecessary
explanations and so on, as to which I need not enter into details.

Lastly, following Heiberg's order, I come to

VII. Interpolated definitions, axioms etc.
Apart from VI. Def. 5 (which may have been interpolated by

Theon although it is found written in the margin of P by the first
hand), the definition of a segment of a circle in Book I. is interpolated,
as is clear from the fact that it occurs in a more appropriate place in
Book III. and Proclus omits it. \l. Def. 2 (reciprocal figures) is rightly
condemned by Simson — perhaps it was taken from Heron — and

i.

i

C9 INTRODUCTION [ch, v

Heiber^ would reject VI I, Def lo, as to which see my note on that
definition. Lastly the double definition of a solid angle (XL Def ii)
constitutes a difficulty. The use of the word iTrt^idvtta suggests that
the first definition may have been older than Euclid, and he may have
quoted it from older iiemtnts^ especially as his own definition which
follows only includes solid angles contained by planes, whereas the m,
other includes other sorts (cf the words ^pap^fio^v, ypafi^<;) which are II
also distinguished by Heron (Def. 24). If the first definition had i
come last, it could have been rejected without hesitation : but it is not 1
so easy to reject the first part up to and including "otherwise"
{aXXm<;). No difficulty need be felt about the definitions of '* oblong,"
"rhombus" "trapezium" and "rhomboid" which are not actually
used in the Elematis; they were no doubt taken from earlier dements
and given for the sake of completeness.

As regards the axioms or, as they are called in the text, common
notions {ttmvai ivvotai), it is to be observed that Proclus says* that
Apollonius tried to prove ''the axioms/' and he gives Apollonius*
attempt to prove Axiom k This shows at all events that Apollonius
had some of the axioms now appearing in the text. But how could
Apollonius have taken a controversial line against Euclid on the
subject of axioms if these axioms had not been Euclid's to his know-
ledge? Andi if they had been interpolated between Euclid's time
and his own, how could Apollonius, living so comparatively short a
time after Euclid, have been ignorant of the fact ? Therefore some of
the axioms are Euclid's (whether he called them common notions, or
axioms^ as is perhaps more likely since Proclus calls them axioms):
and we need not hesitate to accept as genuine the first three discussed
by Proclus, viz. (i) things equal to the same equal to one another,
(2) if equals be added to equals, wholes equal, (5) if equals be
subtracted from equals, remainders equal. The other two mentioned
by Proclus (whole greater than part, and congruent figures equal) are
more doubtful, since they are omitted by Heron, Martianus Capella,
and others. The axiom that "two lines cannot enclose a space" is
however clearly an interpolation due to the fact that I. 4 appeared to
require it The others about equals added to unequals, doubles of
the same thing, and halves of the same thing are also interpolated ;
they are connected with other interpolations, and Proclus clearly
used some source which did not contain them.

Euclid evidently limited his formal axioms to those which seemed
to him most essential and of the widest application ; for he not un-
frequently assumes other things as axiomatic, e.g. in vil. 28 that, if a
number measures two numbers, it measures their difference.

The differences of reading appearing in Proclus suggest the

Question of the comparative purity of the sources used by Proclus,
leron and others, and of our text The omission of the definition of
a s^ment in Book I. and of the old gloss ** which is called the cir-
cumference" in L Def. 15 (also omitted by Heron, Taurus, Sextus

* Pkoehu, pp. 194, loaqq.

CH. v] THE TEXT 63

Empiricus and others) indicates that Proclus had better sources than
we have ; and Heibci^ gives other cases where Proclus omits words
which are in all our MSS. and where Proclus' reading should perhaps
I be preferred. But, except in these instances (where Proclus may have
drawn from some ancient source such as one of the older com-
mentaries), Proclus' MS. does not seem to have been among the best
Often it agrees with our worst MSS., sometimes it agrees with F where
F alone has a certain reading in the text, so that (e.g. in I. 15 Por.)
the common reading of Proclus and F must be rejected, thrice only
does it agree with P alone, sometimes it agrees with P and some
Theonine MSS., and once it agrees with the Theonine MSS. against P
and other sources.

Of the other external sources, those which are older than Theon
generally agree with our best MSS., e.g. Heron, allowing for the
difference in the plan of his definitions and the somewhat free adap-
tation to his purpose of the Euclidean definitions in Books X., XI.

Heiberg concludes that the Elements were most spoiled by inter-
polations about the 3rd c, for Sextus Empiricus had a correct text,
while lamblichus had an interpolated one; but doubtless the purer
text continued for a long time in circulation, as we conclude from the
fact that our MSS. are free from interpolations already found in
lamblichus' MS.

CHAPTER VL

THE SCHOLIA-

» ^ t ,

Heiberg has collected scholia, to the number of about 1500, in
Vol V* of his edition of EucHd, and has also discussed and classified
them in a separate short treatise, in which he added a few others*.

These scholia cannot be regarded as doing much to facilitate the
reading of the Eitntents. As a rule, they contain only such observa-
tions as any intelligent reader could make for himself Among the
few exceptions are XL Nos. 33, 35 (where XL 22^ 23 are extended to
solid angles formed by any number of plane angles), XI L No. 85
(where an assumption tacitly made by Euclid in XIL 17 is proved),
IX. Nos, 28, 29 (where the scholiast has pointed out the error in the
text of JX. 19).

Nor are they very rich in historical information ; they cannot be
compared in this respect with Proclus' commentary on Book I* or
with those of Eutocius on Archimedes and Apollonius, But even
tinder this head they contain some things of interest, e.g. IL No. 1 1
explaining that the gnomon was invented by geometers for the sake of
brevity, and that its name was suggested by an incidental characteristic^
namely that "from it the whole is known {jvmpi^^rai), ^Mh^t of the
whole area or of the remainder, when it (the ywfL^v) is either placed
round or taken away"; iL No. 13, also on the gnomon; IV, No. 2
stating that Book IV. was the discovery of the Pythagoreans ;
V, Na I attributing the content of Book v, to Eudoxus; X. No. i with
its allusion to the discovery of incommensurability by the Pytha-
goreans and to Apollonius* work on irrationals ; X. No. 62 definitely
attributing X, 9 to Theaetetus; XIIL No. i about the "Platonic" figures,
which attributes the cube, the pyramid, and the dodecahedron to the
Pythagoreans, and the octahedron and icosahedron to Theaetetus.

Sometimes the scholia are useful in connexion with the settlement
of the text, (i) directly^ e.g, IIL Na 16 on the interpolation of the
word "within'* (iirri^) in the enunciation of IIL 6, and X, No. 1
alluding to the discussion by "Theon and some others" of irrational
"surfaces" and '* solids," as well as '*lines/' from which we may

^ Heiberg, Om Stka/vrm tU EttkUth Elrminitr^ Kjifbenhft^Q, |S8S, Tbe tr^ct is
wniten is Dankh, but, fortuniLt^ly for those who do not read Danish easilj, Lhe author has
ftprpended (pp. 7^—78} a re&mne in French*

CH.VI] THE SCHOLIA 65

conclude that the scholium at the end of Book x/ is not genuine ;

(2) indirectly in that they sometimes throw light on the connexion
of certain MSS.

Lastly, they have their historical importance as enabling us to
judge of the state of mathematical science at the times when they
were written.

Before passing to the classification of the scholia, Heiberg remarks
that we must separate from them a number of additions in the nature
of scholia which are found in the text of our MSS. but which can, in
one way or another, be proved to be spurious. As they are found
both in P and in the Theonine MSS., they must have been in the MSS.
anterior to Theon (4th c). But they are, in great part, only found in
the margin of P and the Theonine MSS.; in V they are half in the
text and half in the margin. This can hardly be explained except
on the supposition that these additions were originally (in the MSS.
before Theon) in the margin, and that Theon kept them there in his
edition, but that they afterwards found their way gradually into the
text of P as well as of the Theonine MSS., or were omitted altogether,
while particular MSS. have in certain places preserved the old arrange-
ment Of such spurious additions Heiberg enumerates the following:
the axiom about equals subtracted from unequals, the last lines of the
porism to VI. 8, second porisms to V. 19 and to vi. 20, the porism
to III. 31, VI. Def. 5, various additions in Book X., the analyses and
syntheses of XIII. i — 5, and the proposition XIII. 6.

The two first classes of scholia distinguished by Heiberg are
denoted by the convenient abbreviations "SchoL Vat" and "Schol.
Vind."

L Schol. Vat.

It is first necessary to set out the letters by which Heiberg
denotes certain collections of scholia.

P « Scholia in P written by the first hand.

B a Scholia in B by a hand of the same date as the MS. itself,
generally that of Arethas.

F« Scholia in F by the first hand.

Vat = Scholia of the Vatican MS. 204 of the loth c, which has
these scholia on leaves 198 — 205 (the end is missing) as an independent
collection. It does not contain the text of the Ekments,

V*^ = Scholia found on leaves 283 — 292 of V and written in the
same hand as that part of the MS. itself which begins at leaf 235.

Vat 192 =s a Vatican MS. of the 14th c which contains, after
(i) the Elements I. — XIli. (without scholia), (2) the Data with scholia,

(3) Marinus on the Data^ the Schol. Vat as an independent collection
and in their entirety, b^inning with I. No. 88 and ending with Xlll.
No. 44.

The Schol. Vat, the most ancient and important collection of

scholia, comprise those which are found in PBF Vat. and, from vil. 12

. to IX. 1 5, in PB Vat. only, since in that portion of the Elements

F was restored by a later hand without scholia ; they abo include I.

H. £. e .

66 INTRODUCTION [ch. vi

No. 88 which onfy happens to be erased in F, and IX. Nos, 28, 29
which may be left out because F here has a different text In F
and Vat the collection ends with Book X, ; but it must also include
SchoL PB of Books XL^ — xilt, since these are found along with Schol*
Vat to Books L— X* in several MSS. (of which Vat 192 is one) as a
separate collection. The SchoL Vat to Books X,— XIIL are also
found in the collection V^ (where, curiously enough, Xllt Nos. 45, 44
are at the beginning). The Scho!, Vat. accordingly include SchoL
PBV*= Vat 193, and doubtless also those which are found in two of
these sources. The total number of scholia classified by Heiberg as
SchoL Vat is 138,

As r^ards the contents of SchoL Vat Heiberg has the following
observations. The thirteen scholia to Book T. are extracts made
from Proclus by a writer thoroughly conversant with the subject,
and cleverly recast (with some additions). Their author does not
seem to have had the two lacunae which our text of Proclus has
(at the end of the note on t 36 and the beginning of the next note^
and at the beginning of the note on L 43), for the scholia L Nos. 125
and 137 seem to hll the gaps appropriately, at least in part. In
some passages he had better readings than our MSS. have. The rest
of SchoL Vat (on Books ll. — XIII.) are essentially of the same
character as those on Book I., containing prolegomena, remarks on
the object of the propositions, critical remarks on the text, converses,
lemmas; they are, in general, exact and true to tradition. The
reason of the resemblance between them and Proclus appears to be
due to the fact that they have their origin in the commentary of
Pappus, of which we know that Proclus also made use. In support
of the view that Pappus is the source, Heiberg places some of the
SchoL Vat to Book x, side by side with passages from the com-
mentary of Pappus in the Arabic translation discovered by Woepcke* ;
he also refers to the striking confirmation afiforded by the fact that
XIL No. 2 contains the solution of the problem of inscribing in a
given circle a polygon similar to a polygon inscribed in another circle,
which problem Eutocius says' that Pappus gave in his commentary
on the Ekmettts.

But, on the other hand, SchoL Vat. contain some things which
cannot have come from Pappus, e.g. the allusion in X. No, i to Theon
and irrational surfaces and solids, Theon being later than Pappus ;
lit No, 10 about porisms is more like Proclus' treatment of the
subject than Pappus', though one expression recalls that of Pappus
about/ormtn^ (axnM^ri^eaOai) the enunciations of porisms like those
of either theorems or problems.

The Schol. Vat give us important indications as regards the
text of the Elements as Pappus had it In particular, they show that
he could not have had in his text certain of the lemmas in Book x.
For example, three of these are identical with what we find in Schol.

^ Om ScMitrm tU Euklids Elenunttr^ pp. 11, 13 : cf. EukHd'Stutiien^ pp. 170, 171 ;
Woepcke, Mimciru friunt. d tAcad, da Scufues, 1856, xiv. p. 658 sqq.
* Archimedes, ed. Hdbeig, iiL p. 54, 5 — 8.

I

J

(|

; I CH. VI] THE SCHOLIA 67

Vat (the lemma to x. i7 = Schol. x. No. 106, and the lemmas to
• X. 54, 60 come in Schol. X. No. 328) ; and it is not possible to suppose
* \ that these lemmas, if they were already in the text, would also be
given as scholia. Of these three lemmas, that before X. 60 has
already been condemned for other reasons; the other two, un-
objectionable in themselves, must be rejected on the ground now
stated. There were four others against which Heiberg found nothing
to urge when writing his prolegomena to Vol. v., viz. the lemmas
before X. 42, x. 14, x. 22 and X. 33. Of these, the lemma to X. 22
is not reconcilable with Schol. X. No. 161, which takes up the
assumption in the text of Eucl. x. 22 as if no lemma had gone before.
The lemma to X. 42, which, on account of the words introducing it
(see p. 60 above), Heiberg at first hesitated to regard as an inter-
polation, is identical with Schol. X. No. 270. It is true that in
Schol. X. No. 269 we find the words "this lemma has been proved
before (iv roh ifiirpoaOev), but it shall also be proved now for
convenience' sake (rov iroifiov tv€KaY and it is possible to suppose
that " before " may mean in Euclid's text before X. 42 ; but a proof
in that place would surely have been as ** convenient " as could be
desired, and it is therefore more probable that the proof had been
given by Pappus in some earlier place. (It may be added that the
lemma to x. 14, which is identical with the lemma to XI. 23, con-
demned on other grounds, is for that reason open to suspicion.)

Heiberg's conclusion is that all the lemmas are spurious, and that
most or all of them have found their way into the text from Pappus'
commentary, though at a time anterior to Theon's edition, since
they are found in all our MSS. This enables us to fix a date for these
interpolations, namely the first half of the 4th c.

Of course Pappus had not in his text the interpolations which,
from the fact of their appearing only in some of our MSS., are seen to
be later than those above-mentioned. Such are the lemmas which
are found in the text of V only after x. 29 and x. 31 respectively and
are given in Heiberg's Appendix to Book x. (numbered 10 and 11).
On the other hand it appears from Woepcke's tract^ that Pappus
already had X. 115 in his text : though it does not follow from this
that the proposition is genuine but only that interpolations began
very early.

Theon interpolated a proposition (or lemma) between X. 12 and
X. 13 (No. 5 in Heiberg*s Appendix). Schol. Vat. has the same
thing (X. No. 125). The writer of the scholia therefore did not find
this lemma in the text. Schol. Vat. IX. Nos. 28, 29 show that neither
did he find in his text the alterations which Theon made in Eucl. IX.
19; the scholia in fact only agree with the text of P, not with Theon's.
This suggests that Schol. Vat. were written for use with a MS. of the
ante-Theonine recension such as P is. This probability is further
confirmed by a certain independence which P shows in several places
when compared with the Theonine MSS. Not only has P better
readings in some passages, but more substantial divergences, and,

* Woepcke, op, cit. p. 70a.

5—2

».i.

» introi>ucti6n

in particularp the absence in P of three notes of a historical character
which are added ^ wholly or partly from Proclus, in the Theonine MSS,
attests an independent and more primitive point of view in P,

In view of the distinctive character of P, it is possible that some
of the scholia found in it in the first hand, but not in the other
sources of SchoK Vat, also belong to that collection ; and several
circumstances confirm this. Schol, XI I L No. 45. found in P only,
which relates to a passage in Eucl Xill. 15, shows that certain words
in the text, though older than Theon» are interpolated ; and, as the
scholium is itself older than Theon, is headed *' tkird lemma/' and
follows a "second lemma" relating to a passage in the text im-
mediately preceding, which ** second lemma" belongs to Schol Vat
and is taken from Pappus, the *'third" in all probability came from
Pappus also. The same is true of SchoL XlL No. 72 and XII L No* 69,
which are respectively identical with the propositions t^u/^o XL 38
(Heiberg, App, to Book XL, No. 5) and xlll. 6; for both of these
interpolations are older than Theon, Moreover most of the scholia
which P in the first hand alone has are of the same character as
Schoh Vat Thus VIL No. 7 and XIIL No. i introducing Books vil.
and XIIL respectively are of the same historical character as several
of SchoL Vat, ; that Vll, No. 7 appears in the t€zt of P at the
beginning of Book vil constitutes no difficulty. There are a number
of converses^ remarks on the relation of propositions to one another,
explanations such as XIL No. 89 in which it is remarked that 4>, H
in Euclid's figure to XIL 17 {Z^ V in my figure) are really the same
point but that this makes no difference in the proof. Two other
Schol. P on XIL 17 are connected by their headings with xn. No. 72
mentioned above, XL No, 10 (P) is only another form of XL
No, 1 1 (B) ; and B often, alone with P, has preserved Schol. Vat
On the whole Heiberg considers some 40 scholia found in P alone to
belong to Schol. Vat

The history of Schol, Vat. appears to have been, in its main
outlines, the following. They were put together after 500 A<D., since
they contain extracts from Proclus, to which we ought not to assign
a date too near to that of Proclus' work itself; and they must at least
be earlier than the latter half of the 9th c, in which B was written.
As there must evidently have been several intermediate links between
the archetype and B, we must assign them rather to the first half of
the period between the two dates, and it is not improbable that they
were a new product of the great development of mathematical studies
at the end of the 6th c. (Isidorus of Miletus). The author extracted
what he found of interest in the commentary of Proclus on Book L
and in that of Pappus on the rest of the work, and put these extracts
in the margin of a MS, of the class of P, As there are no scholia to
L I — 22, the first leaves of the archetype or of one of the eariiest
copies must have been lost at an early date, and it was from that
mutilated copy that partly P and partly a MS. of the Theonine class
were taken, the scholia being put in the margin in both. Then the
collection spread through the Theonine MSS., gradually losing some

CH.VI] THE SCHOLIA 69

I ca.Y

I scholia which could not be read or understood, or which were

I accidentally or deliberately omitted. Next it was extracted from

\ one of these MSS. and made into a separate work which has been

I preserved, in part, in its entirety (Vat 192 etc.) and, in part, divided

I into sections, so that the scholia to Books X. — XIII. were detached

; (V^). It had the same fate in the MSS. which kept the original

arrangement (in the mai^in), and in consequence there are some MSS.

where the scholia to the stereometric Books are missing, those Books

having come to be less read in the period of decadence. It is from

one of these MSS. that the collection was extracted as a separate work

such as we find it in Vat. (loth c).

II. The second great division of the scholia is SchoL Vind.

This title is taken from the Viennese MS. (V), and the letters used
by Heiberg to indicate the sources here in question are as follows.

V* = scholia in V written by the same hand that copied the MS.
itself from fol. 235 onward.

q=s scholia of the Paris MS. 2344 (q) written by the first hand.

1 = scholia of the Florence MS. Laurent XXVlii, 2 written in the
13th-- 14th c, mostly in the first hand, but partly in two later
hands.

V^s= scholia in V written by the same hand as the first part
(leaves i— 183) of the MS. itself; V^ wrote his scholia after V*.

q* = scholia of the Paris MS. (q) found here and there in another
hand of early date.

Schol. Vind. include scholia found in VH). 1 is nearly related to
q ; and in fact the three MSS. which, so far as Euclid's text is con-
cerned, show no direct interdependence, are, as regards their scholia,
derived from one original. Heiberg proves this by reference to the
readings of the three in two passages (found in Schol. I. No. 109 and
X. No. 39 respectively). The common source must have contained,
besides die scholia found in the three MSS. V*ql, those also which
are contained in two of them, for it is more unlikely that two of the
three should contain common interpolations than that a particular
scholium should drop out of one of them. Besides V* and q, the
scholia V^ and q' must equally be referred to Schol. Vind., since the
greater part of their scholia are found in 1. There is a lacuna in q
from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements
Schol. Vind. are represented by VI only. Heiberg pves about 450
numbers in all as belonging to this collection.

Schol. Vind. did not all come from one source ; this is shown by
diflferences of substance, e.g. between x. Nos. 36 and 39, and by
differences of time of writing : e.g. VI. No. 52 refers at the b^inning
to No. 55 with the words "as the scholium has it" and is therefore
later than that scholium ; X. No. 247 is also later than X. No. 246.

The scholia to Book I. are here also extracts from Proclus, but
more copious and more verbatim than in Schol. Vat The author
has not always understood Proclus; and he had a text as bad as
that of our MSS., with the same lacunae. The scholia to the other

INTRODUCTION ^> [ch. vi

Mil*' *■

3^f*;

Books are partly drawn (r) from Schol Vat, the MSS. representing
Schol Vind. and Schol. Vat in these cases showing nearly all possible
combinations; but there is no certain trace in Schol Vind< of the
scholia peculiar to P. The author used a copy of Schol Vat, in the
form in which they were attached to the Theonine text ; thus Schol
Vind. correspond to BF Vat, where these diverge from P, and
especially closely to B, Besides Schol Vat., the editors of Schol
Vind. used (2) other old collections of scholia of which we find traces
in B and F; Schol Vind, have also some scholia common with b.
The scholia which Schol Vind, have in common with BF come from
two different sources, and were apparently afterwards introduced
into the other MSS. ; one result of this is that several scholia are
reproduced twice.

Butj besides the scholia derived from these sources, Schol Vind
contain a large number of others of late date, characterised by in-
correct language or by triviality of content (there are many examples
in numbers, citations of propositions used, absurd diropiai, and the
hke). Unlike Schol Vat, these scholia often quote words from Euclid
as a heading (in one case a heading is inserted in Schol Vind. where
a scholium without the heading is quoted from Schol Vat, see V.
No. 14). The explanations given often presuppose very little know-
ledge on the part of the reader and frequently contain obscurities
and gross errors.

Schol Vind* were collected for use with a US. of the Theonine
class; this follows from the fact that they contain a note on the
proposition vuigo VII. 22 interpolated by Theon (given in Heiberg's
A pp. to Vol II. p. 430). Since the scholium to vil 39 given in V and
p in the text after the title of Book VIIL quotes the proposition as
VII. 39, it follows that this scholium must have been written before
the interpolation of the two propositions vu^a VII. 20, 22 ; Schol
Vind. contain (viL No* 80) the first sentence of it, but without the
heading referring to VI l* 39* Schol VIL No* 97 quotes vlL 33 as
vn* 34, so that the proposition vui^ VII. 22 may have stood in the
scholiast's text but not the later interpolation vu/go VIL 20 (later
because only found in B in the margin by the first hand). Of course
the scholiast had also the interpolations earlier than Theon*

For the date of the collection we have a lower limit in the date
(12th c.) of MSS. in which the scholia appear* That it was not much
earlier than the 1 2th c. is indicated ( r ) by the poverty of its contents,
(2) by the quality of the MS. of Proclus which was used in the
compilation of it (the Munich MS. used by Friedlein with which the
scholiast's excerpts are essentially in agreement belongs to the i ith —
t2th c.)^ (3) by the fact that Schol Vind. appear only in uss. of the
1 2th c* and no trace of them is found in our MSS, belonging to
the 9th — loth c* in which Schol Vat are found. The collection may
therefore probably be assigned to the iith c. Perhaps it may be in
part due to Psellus who lived towards the end of that century : for in
a Florence MS. (Magltabecch. Xl^ 53 of the 15th c) containing a
mathematical compendium intended for use in the reading of Aristotle

CH. VI] THE SCHOLIA 71

the scholia I. Nos. 40 and 49 appear with the name of Psellus
attached.

f.* Schol. Vind. are not found without the admixture of foreign

M elements in any of our three sources. In 1 there are only very few
such in the first hand. In q there are several new scholia in the first
hand, for the most part due to the copyist himself The collection of
scholia on Book X. in q (Heiberg's q^) is also in the first hand ; it is
not original, and it may perhaps be due to Psellus (Maglb. has some
definitions of Book X. with a heading "scholia of... Michael Psellus
on the definitions of Euclid's loth Element'' and Schol. x. No. 9),
whose name must have been attached to it in the common source of
Maglb. and q ; to a great extent it consists of extracts from Schol.
Vind. taken from the same source as VI. The scholia q* (in an
ancient hand in q), confined to Book II., partly belong to Schol. Vind.
and partly correspond to W (Bologna MS.), q* and q^ are in one hand
(Theodorus Antiochita), the nearest to the first hand of q ; they are
doubtless due to an early possessor of the MS. of whom we know
nothing more.

V* has, besides Schol. Vind., a number of scholia which also appear
in other MSS., one in BFb, some others in P, and some in v (Codex
Vat 1038, 13th c.) ; these scholia wei-e taken from a source in which
many abbreviations were used, as they were often misunderstood by
V". Other scholia in V* which are hot found in the older sources —
some appearing in V* alone — are also not original, as is proved by
mistakes or corruptions which they contain ; some others may be due
to the copyist himself.

V^ seldom has scholia common with the other older sources ; for
the most part they either appear in V* alone or only in the later
sources as v or P (later scholia in F), some being original, others not
In Book x. V^ has three series of numerical examples, (i) with Greek
numerals, (2) alternatives added later, also mostly with Greek numerals,
(3) with Arabic numerals. The last class were probably the work of
the copyist himself. As V^ belongs to the same time as the MS.
(i2th c), these examples give an idea of the facility with which
calculations were made in Byzantium at that time. They show too
that the Greek method of writing numbers still preponderated in the
nth c, but that the use of the Arabic numerals (in the East- Arabian
form) was thoroughly established in the 12th c

Of collections in other hands in V distinguished by Heiberg (see
preface to Vol. v.), V* has very few scholia which are found in other
sources, the greater part being original ; V*, V* are the work of the
copyist himself; V^ are so in part only, and contain several scholia
from Schol. Vat. and other sources. V* and V* are later than 13th
— 14th c, since they are not found in f (cod. Laurent. XXVIII, 6) which
was copied from V and contains, besides V* V^ the greater part of
V^ and VI. No. 20 of V* (in the text).

In P there are, besides P* (a quite late hand, probably one of the
old Scriptores Graeci at the Vatican), two late hands (P*), one of
which has some new and independent scholia, while the other has

72 INTRODUCTION

added the greater part of Schol Vm±, partly in the margin and
partly on pieces of leaves stitched on,

Our sources for Schol Vat also contain other elements. In P
there were introduced a certain number of extracts from Proclus, to
supplement Schol. Vat, to Book l. ; they are all written with a
different ink from that used for the oldest part of the MS,, and the
text is inferior. There are additions in the other sources of Schol
Vat (F and B) which point to a common source for FB and which
are nearly all found in other MSa, and, in particular, in Schol Vind.,
which also used the same source ; that they are not assignable to
Schol Vat results only from their not being found in Vat Of other
additions in F, some are peculiar to F and some common to it and b;
but they are not original F* (scholia in a later hand in F) contains
three original scholia ; the rest come from V. B contains, besides
scholia common to it and F, b or other sources, several scholia which
seem to have been put together by Arethas, who wrote at least a part
of them with his own hand.

Heiberg has satisfied himself, by a closer study of b, that the
scholia which he denotes by b, fi and b* are by one hand ; they are
mostly to be found in other sources as well though some are original
By the same hand (Theodorus Cabasilas, 1 5th c.) are also the scholia
denoted by W, B*, b* and B*. These scholia come in great part from
Schol Vind., and in making these extracts Theodorus probably used
one of our sources, 1, mistakes in which often correspond to those of
Theodorus, To one scholium is attached the name of Demetrius (who
must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c*);
but it could not have been written by him, since it appears in B and
Schol Vind. Nor are all the scholia which bear the name of
Theodorus due to Theodorus himself, though some are so.

As B* (a late hand in B) contains several of the original scholia of
b", B* must have used b itself as his source, and, as all the scholia in
B* are in b, the latter is also the source of the scholia in B* which are
found in other MSS. B and b were therefore, in the 1 5th c, in the
hands of the same person ; this explains, too, the fact that b in a late
hand has some scholia which can only come from B, We arrive then
at the conclusion that Theodorus Cabasilas, in the 15th c, owned both
the MSS. B and b, and that he transferred to B scholia which he had
before written in b, either independently or after other sources, and
inversely transferred some scholia from B to b. Further, B' are
earlier than Theodorus Cabasilas, who certainly himself wrote B* as
well as b* and b*.

An author's name is also attached to the scholia VI. No. 6 and
X. No. 223, which are attributed to Maximus Planudes (end of 13th c)
along with scholia on I. 31, x. 14 and x. 18 found in 1 in a quite late
hand and published on pp. 46, 47 of Heiberg's dissertation. These
seem to have been taken from lectures of Planudes on the Elements
by a pupil who used 1 as his copy.

There are also in 1 two other byzantine scholia, written by a late
hand, and bearing the names loannes and Pediasimus respectively ;

r

t

CH.VI] THE SCHOLIA 73

these must in like manner have been written by a pupil after lectures
of loannes Pediasimus (first half of 14th c), and this pupil must also
have used 1.

Before these scholia were edited by Heiberg, very few of them had
been published in the original Greek. The Basel editio princeps has a
few (v. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken,
some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others
probably from the Venice MS. (Marc. 301) also used by him; one
published by Heiberg, not in his edition of Euclid but in his paper
on the scholia, may also be from Venet. 301, but appears also in
Paris. Gr. 2342. The scholia in the Basel edition passed into the
Oxford edition in the text, and were also given by August in the
* Appendix to his Vol. II.

I Several specimens of the two series of scholia (Vat. and Vind.)

' were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.)
I and by KnocYit {Untersuchungen fiber die neu aufgefundenen Scholien
des Proklus, Herford, 1865).

The scholia published in Latin were much more numerous. G.
Valla {De expetettdis et fugiendis rebus^ 1501) reproduced apparently
some 200 of the scholia included in Heiberg's edition. Several of
these he obtained from two Modena MSS. which at one time were
in his possession (Mutin. Ill B, 4 and II E, 9, both of the 15th c);
but he must have used another source as well, containing extracts
from other series of scholia, notably Schol. Vind. with which he has
some 87 scholia in common. He has also several that are new.

Commandinus included in his translation under the title *' Scholia
antiqua" the greater part of the Schol. Vat. which he certainly
obtained from a MS. of the class of Vat. 192; on the whole he
adhered closely to the Greek text Besides these scholia Com-
mandinus has the scholia and lemmas which he found in the Basel
editio princeps^ and also three other scholia not belonging to Schol.
Vat, as well as one new scholium (to xil. 13) not included in
Heiberg's edition, which are distinguished by different type and were
doubtless taken from the Greek MS. used by him along with the
Basel edition.

In Conrad Dasypodius' Lexicon mathematicum published in 1573
there is (on fol. 42 — 44) "Graecum scholion in definitiones Euclidis
libri quinti elementorum appendicis loco propter pagellas vacantes
annexum." This contains four scholia, and part of two others,
published in Heiberg's edition, with some variations of readings, and
with some new matter added (for which see pp. 64 — 6 of Heiberg's
pamphlet). The source of these scholia is revealed to us by another
work of Dasypodius, Isaaci Monachi Scholia in Euclidis elententorum
geometriae sex priores libros per C Dasypodium in latinutn sermonem
translata et in lucent edita (1579). This work contains, besides
excerpts from Proclus on Book I. (in part closely related to Schol.
Vind.), some 30 scholia included in Heiberg's edition, several new
scholia, and the above-mentioned scholia to the definitions of Book V.
published in Greek in 1573. After the scholia follow ** Isaaci Monachi

74 INTRODUCTION [c», vr

prolegomena in Eudidis Elementorum geometriae Hbros" (two
definitions of geometry) and "Van a miscellanea ad geometriae cogni-
tionem necessana ab Isaaco Monacho collecta" (mostly the same as
pp, 252^ 24 — 272, 27 in the Variae Colkcticnes included in Hultsch's
Heron); lastly, a note of Dasypodius to the reader says that these
schoha were taken '^ex clarissimi vrri Joan n is Sambuci anttquo cod ice
many propria Isaaci Monachj scripto/* Isaak Monachus is doubtless
Isaak Argyrus, 14th c; and Dasypodtus used a MS. in which, besides
the passage in Hultsch's Variae CQikctwfus, were a number of
scholia marked in the margin with the name of Isaak (c£ those in b
under the name of Theodorus Cabasilas). Whether the new schoJia
are original cannot be decided until they are published in Greek ; but
it is not improbable that they are at all events independent arrange-
ments of older scholia. All but five of the others, and all but one of
the Greek scholia to Book v.^ are taken from Schol. Vat ; three of the
excepted ones are from Schol, Vind,, and the other three seem to
come from F (where some words of them are illegiblej but can be
supplied by means of Mut in B, 4, which has these three scholia and
generally shows a certain likeness to Isaak s scholia).

Dasypodius also published in 1564 the arithmetical commentary
of Barlaam the monk (14th c) on Eucl Book ll., which finds a place
in Appendix IV, to the Scholia in Heiberg s edition. -

CHAPTER VII.

EUCLID IN ARABIA.

We are told by Hajl Khalfa* that the Caliph al-Mansur (754-775)
I sent a mission to the Byzantine Emperor as the result of which he

(obtained from him a copy of Euclid among other Greek books, and
again that the Caliph al-Ma'mun (813-833) obtained manuscripts of
* Euclid, among others, from the Byzantines. The version of the
} Elements by al-Hajjaj b. Yusuf b. Matar is, if not the very first, at
least one of the first books translated from the Greek into Arabic*.
According to the Fihrist^ it was translated by al-Hajjaj twice; the
first translation was known as " Haruni " (" for Harun "), the second
I bore the name "Ma'muni" ("for al-Ma'mun") and was the more trust-
worthy. Six Books of the second of these versions survive in a Leiden
MS. (Codex Leidensis 399, i) which is being published by Besthom
and Heiberg*. In the preface to this MS. it is stated that, in the reign
of Harun ar-RashId (786-809), al-Hajjaj was commanded by Yahya
! b. Khalid b. Barmak to translate the book into Arabic Then, when
^ al-Ma'mun became Caliph, as he was devoted to learning, al-Hajjaj
saw that he would secure the favour of al-Ma'mun "if he illustrated and
'. expounded this book and reduced it to smaller dimensions. He
I' accordingly left out the superfluities, filled up the gaps, corrected or
' removed the errors, until he had gone through the bcK)k and reduced
it, when corrected and explained, to smaller dimensions, as in this
I copy, but without altering the substance, for the use of men endowed

• with ability and devoted to learning, the earlier edition, being left in
'• the hands of readers."

The FUifist goes on to say that the work was next translated by
Ishaq b. Hunain, and that this translation was improved by Thabit b.
Qurra. This Abu Ya'qub Ishaq b. Hunain b. Ishaq al-'ibadi (d. 910)
was the son of the most famous of Arabic translators, Hunain b. Ishaq
al-'ibadi (809-873), a Christian and physician to the Caliph al-
Mutawakkil (847-861). There seems to be no doubt that Ishaq, who

^ Lexicon bibliogr. et encychp, ed. FlUgel, III. pp. 91, 93.

• Klamroth, Zeitschrift dtr Deutschen Morgenldndischen GeseUschafU XXXV. p. 303.
' » Fihrist (tr. Sutcr), p. 16.

^ Codtx Leidensis 399, i . Euclidis EUmenta ex interfretaiume al'Hadschdschadschii cum

• ccmmeniariis al'NarizU^ Hauniae, part I. i. 1893, part I. ii. 1897, part II. i. 1905.

i'

r6 INTRODUCTION [ra. vu

must have known Greek as well as hts father, made his translation
direct from the Greek, The revision must apparently have been the
subject of an arrangement between Ishaq and T ha bit, as the latter
died in 90 ! or nine years before Ishaq, Thabit undoubtedly consulted
Greek MSS. for the purposes of his revision. This is expressly stated
in a marginal note to a Hebrew version of the Elements^ made from
Ishaq's, attributed to one of two scholars belonging to the same family ^
viz, either to Moses b. Tibbon (about 1244-1274) or to Jakob b, Machir
(who died soon after 1306)^ Moreover Thabit observes, on the pro-
position which he gives as ix. 3 1, that he had not found this proposition
and the one before it in the Greek but only in the Arabic ; from which
statement Klamroth draws two conclusions, ( 1 ) that the Arabs had
already begun to interest themselves in the authenticity of the text
and (2) that Thabit did not alter the numbers of the propositions in
Ishaq's translation'. The Fihrisi also says that Yuhanna al-Qass (i.e.
" the Priest ") had seen in the Greek copy in his possession the pro-
position in Book I. which Thabit took credit for, and that this was
confirmed by Nazlf, the physician, to whom Yuhanna had shown it
This proposition may have been wanting in Ishaq, and Thabit may
have added it, but without claiming it as his own discovery'. As
a fact, I. 45 is missing in the translation by al-Hajjaj,

The original version of Ishaq wtthcut the improvements by ThSbit
has probably not survived any more than the first of the two versions
by al'Hajjaj ; the divergences between the MSS. are apparently due to
the voluntary or involuntary changes of copyists, the former class
varying according to the degree of mathematical knowledge possessed
by the copyists and the extent to which they were influenced by
considerations of practical utility for teaching purposes*. Two MSS* '
of the Ishaq -Thabit version exist in the Bodleian Library (No. 279
belonging to the year 1238, and No. 280 written in i26o~i)"; Books
L — XIIL are in the Ishaq-Thabit version, the non-Euclidean Books
XIV,, XV. in the translation of Qusta b. Luqa al-Ba'labakkl (d. about
912)* The first of these MSS. (No- 279) is that (O) used by Klamroth
for the purpose of his paper on the Arabian Euclid. The other MS.
used by Klamroth is (K) Kj^enhavn LXXXI, undated but probably
of the 13th c, containing Books V. — XV., Books v. — X. being in the
Ishaq-Thabit version, Books XL — XIII. purporting to be in al-Hajjaj's
translation, and Books xiv., XV. in the version of Qusta b. Luqa. In
not a few propositions K and O show not the slightest difference, and,
even where the proofs show considerable diflferences, they are generally
such that, by a careful comparison, it is possible to reconstruct the
common archetype, so that it is fairly clear that we have in these cases,
not two recensions of one translation, but arbitrarily altered and

^ Steinfchneider, ZeiUchrift fiir Math, u, Pkysik, xxxi., hist-litt Abtheilung, pp. 85,

"Klamroth, p. 379. * Stdmchndder, p. 88.

^ Klamroth, p. 300.

* These mss. are described by NicoU and Pusey, Caiaiogus cod, mss, orimi. diN. Bod-
Umttoi, pt IL 1835 (pp. 357 — 262).

I

CH. vii] EUCLID IN ARABIA 77

shortened copies of one and the same recension ^ <i^The Bodleian MS.
No. 280 contains a preface, translated by Nicoll, which cannot be by
Thabit himself because it mentions Avicenna (980-1037) and other
later authors. The MS. was written at Mara^a in the year 1 260-1 and
. has in the margin readings and emendations from the edition of
f Naslraddin at-TusI (shortly to be mentioned) who was living at Maraga
( at the time. Is it possible that at-TusI himself is the author of the
I preface'? Be this as it may, the preface is interesting because it
throws light on the liberties which the Arabians allowed themselves
^ to take with the text. After the observation that the book (in spite
r of the labours of many editors) is not free from errors, obscurities,
L redundancies, omissions etc, and is without certain definitions neces-
I sary for the proofs, it goes on to say that the man has not yet been
' found who could make it perfect, and next proceeds to explain
(i) that Avicenna *'cut out postulates and many definitions" and
r attempted to clear up difficult and obscure passages, (2) that Abu'l
I Wafa al-Buzjanl (939-997) "introduced unnecessary additions and
I left out many things of great importance and entirely necessary,"
inasmuch as he was too long in various places in Book VI. and too
short in Book X. where he left out entirely the proofs of the a^tamae,
while he made an unsuccessful attempt to emend XII. 14, (3) that Abu
Ja'far al-Khazin (d. between 961 and 971) arranged the postulates
excellently but " disturbed the number and order of the propositions,
reduced several propositions to one " etc. Next the preface describes
the editor's own claims* and then ends with the sentences, " But we
have kept to the order of the books and propositions in the woric itself
(i.e. Euclid's) except in the twelfth and thirteenth books. For we have
dealt in Book xiil. with the (solid) bodies and in Book Xll. with the
surfaces by themselves."

After Thabit the Fihrist mentions Abu 'Uthman ad-Dimashq! as
having translated some Books of the Elements including Book X. (It
is Abu 'Uthman's translation of Pappus* commentary on Book x.
which Woepcke discovered at Paris.) The Fihrist adds also that
" Nazlf the physician told me that he had seen the tenth Book of
Euclid in Greek, that it had 40 propositions more than the version
in common circulation which had 109 propositions, and that he had
determined to translate it into Arabic."

But the third form of the Arabian Euclid actually accessible to us
is the edition of Abu Ja'far Muh. b. Muh. b. al-Hasan Naaraddin
at-TusI (whom we shall call at-TusI for short), bom at Tus (in
Khurasan) in 1201 (d. 1274). This edition appeared in two forms, a
larger and a smaller. The larger is said to survive in Florence only
(Pal. 272 and 313, the latter MS. containing only six Books) ; this was
published at Rome in 1 594, and, remarkably enough, some copies of

' Klamroth, pp. 306—8.

* Steinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitichrift
. , fiir Math, u. Physik, XXIX., hist.-Iitt. AUh. p. 16.

* This seems to include a rearrangement of the contents of Books xiv., xv. added to the
t Elements,

> •

V-

^ ...^

* Cnrtse, 0^. ^. p. so; Heibeig, EukUd-Studim^ p. 178.

• Hdberg^s Eudidy voL v. p. ci.

*• Klamrotb, pp. 173— 4.

ft INTRODUCTIOli^ [cb. ¥ii ,

this edition are to be found with 12 and some with 13 Books, some
with a Latin title and some without\ But the book was printed tn
Arabic, so that K^stner remarks that he will say as much about it as
can be said about a book which one cannot read* The shorter form,
which however, in most MSS., is in 15 Books, survives at Berlin, Munich,
Oxford, British Museum (974, 1334*, 1335X P^ns (2465. 2466), India
Office, and Constantinople ; it was printed at Constantinople in
1801, and the first six Books at Calcutta in i824\

At*Tusl's work is however not a translation of Euclid's text, but a
re*wr!tten Euclid based on the older Arabic translations. In this
respect it seems to be like the Latin version of the EUmt^nts by
Campanus {Cam pan o), which was first published by Erhard Ratdolt
at Venice in J 482 (the first printed edition of Euclid*). Campanus
(13th c.) was a mathematician^ and it is likely enough that he allowed
himself the same liberty as at-TusI in reproducing Euclid. What-
ever may be the relation between Campanus' version and that of
Athelhard of Bath (about 1 120), and whether, as Curt^e thinks'^ they
both used one and the same Latin version of loth — 1 Ith c, or whether
Campanus used Athclhard's version in the same way as at-Tusi used
those of his predecessors', it is certain that both versions came from
an Arabian source, as is evident from the occurrence of Arabic words
in them I Campanus' version is not of much service for the purpose
of forming a judgment on the relative authenticity of the Greek and
Arabian tradition ; but it sometimes preserves traces of the purer
source, as when it omits Theons addition to wu 33'* A curious
circumstance is that, while Campanus' version agrees with at-Tust's
in the number of the propositions in all the genuine Euclidean Bcwks
except V. and IX., it agrees with Athelhard's in having 34 propositions i
in Book V. (as against 25 in other versions), which confirms the view ;
that the two are not independent, and also leads, as Klamroth says,
to this dilemma: either the additions to Book v. are Athelhard's
own, or he used an Arabian Euclid which is not known to us^.
Heiberg also notes that Campanus' Books xiv., xv. show a certain
agreement with the preface to the Thabit-Ishaq version, in which the
author claims to have (1) given a method of inscribing spheres in the
five r^ular solids, (2) carried further the solution of the problem how

' Sater, Die Mathematikir und Astronomm der Arader, p. 151. The Latin title is
Eiulidis iUmeniorum geomeirkorum Wnri tredeeim. Ex traJitione doctissimi Nasiridim I
Tktmi nunc frimnm arabice impressu Romae in typographia Medicea MDXCiv. Cum
lioentia sapenonim. {

* KMstner, Geschukie der Maihemaiik^ I. p. 367. I

* Smer has a note that this MS. is very old, having been copied from the original in the y
anthor's lifetime. '

* Suter, p. 151.

* Described by Kistner, Geschickte der Mathemaiik^ i. pp. S89--399, and by Weiss-
enbom, Die Ubersettungm da EukHd durch Campano und Zamberti^ Halle a. S., 188s,
pp. I— -7. See also in/rat Chapter vni, p. 97.

* Sonderabdruck des Jakresberichies iiSer die Fortschritte der kiassiuhen AUerthuwn-
wissenschfai vom Oki. 1879—1881, Berlin, 1884.

' Klamroth, p. 371.

CH. vii] EUCLID IN ARABIA 79

to inscribe any one of the solids in any other and (3) noted the cases
where this could not be done\

With a view to arriving at what may be called a common measure
of the Arabian tradition, it is necessary to compare, in the first place,
the numbers of propositions in the various Books. Hajl Khalfa says
that al-Hajjaj's translation contained 468 propositions, and Thabit's
478 ; this is stated on the authority o( at-Tusi, whose own edition
contained 468*. The fact that Thabit's version had 478 propositions
is confirmed by an index in the Bodleian MS. 279 (called O by
Klamroth). A register at the beginning of the Codex Leidensis 399, i
which gives Ishaq's numbers (although the translation is that of
al-Hajjaj) apparently makes the total 479 propositions (the number in
Book XIV. being apparently 1 1, instead of the 10 of O*). I subjoin a

1

table ot relative nu
the corresponding

mbers tal
numbers

<en trom Kl
in August's

amroth, to which I have ad
and Heiberg*s editions of

1

Greek text.

-

1

The Arabian Euclid

The Greek Euclid

Books

Ishaq

at-TusI

Campanus

Gregory

August

Heiberg

1

I

48

48

48

48

48

48

II

14

14

14

14

14

14

•

III

36

36

36

37

37

37

IV

16

16

16

16

16

16

V

25

25

34

25

25

25

VI

33

32

32

33

33

33

VII

39

39

39

41

41

39

VIII

27

25

25

27

. ^7

27

IX

38

36

39

36

36

36

X

109

107

107

117

116

"5

XI

41

41

41

40

40

39

XII

15

IS

15

18

18

18

XIII

21

18

18

18

18

18

462

452

464

470

469

465

[xiv

10

10

18

7

?

XV

6

6

13

10

478

468

495

487

?]

The numbers in the case of Heiberg include all propositions which
he has printed in the text; they include therefore XIII. 6 and III. 12
now to be regarded as spurious, and X. 112 — 115 which he brackets
as doubtful. He does not number the propositions in Books XIV., xv.,
but I conclude that the numbers in P reach at least 9 in XIV., and 9
in XV.

> Heiberg, Zeitichrift fur Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. i\,

• Klamroth, p. 274; Steinschneider, Zeitichrift fiir Math, u, Physti, xxxi., hist.-litt.
Abth. p. 98.

* Besthom- Heiberg read ** 11 ?" asihe number, Klamroth had read it as 31 (p. 373).

1

80 INTRODUCTION [ch. to

The Fihrist confirms the number 109 for Book X,, from which
Klamroth concludes that Ishaq's version was considered as by far the
most authoritative.

In the text of O, Book iv, consists of 17 propositions and Book
XIV, of 1 2, differing in this respect from its own table of contents ; IV.
15, 16 in O are really two proofs of the same proposition.

In al-Hajjaj's version Book L consists of 47 propositions only, 1, 45
being omitted. It has also one proposition fewer in Book 11 1., the
Heronic proposition ill. 12 being no doubt omitted*

In speaking of particular propositions, I shall use Heiberg's
numbering, except where otherwise stated.

The difference of 10 propositions between Thabit*Ishaq and
at-Tusi 15 accounted for thus :

(i) The three propositions Vl 12 and x. 28, 29 which both Ishaq
and the Greek text have are omitted in at-Tusi.

(2) Ishaq divides each of the propositions Xlll. i — 3 into two,
making six instead of three in at-Tus! and in the GreeL

(3) Ishaq has four propositions (numbered by him viii. 24^ 25,

IX. 30. 31) which are neither in the Greek Euclid nor in at*TusL
Apart from the above differences aUHajjaj {so far as we know),

Ishaq and at-Tusl agree \ but their Euclid shows many differences
from our Greek text. These differences we will classify as follows ^

K Propositions.

The Arabian Euclid omits VI L 20, 22 of Gregory's and August's
editions (Heiberg, App. to Vol Ih pp. 428«32) ; VIIL 16, 17 ; X. 7, 8,
13, 16, 24, 112, 113, 114, besides a lemma vtdgo X. 13, the proposition

X. 117 of Gregory's edition, and the scholium at the end of the Book
(see for these Heiberg*s Appendix to Vol. in, pp. 382, 408 — 416);
XL 38 in Gregory- and August (Heibcrg, App. to Vol, Iv, p. 354);
XII. 6, 13, 14; {also all but the first third of Book XV.).

The Arabian Euclid makes II L 11, 12 into one proposition, and
divides some propositions {X. 31, 32 ; XL 31, 34; xlll. i — 3) into two
each.

The order is also changed in the Arabic to the following extent
V, 12, 13 are interchanged and the order in Books VI., VIL, IX. —
xiiL is :

VL 1—8, 13, II, 12. % 10, 14— *7» ift 20, 18, 21, 22| 24, 26, 23,
25,27—30,32,31,33.

VI I. I— 2Q, 22, 21, 23—28, 31, 32i 29, 30, 33 — 39.

IX. 1-^13, 20, 14 — 19, 21 — 25, 27, 26, 28 — 36, with two new pro-
positions coming before prop. 30.

X. 1—6, 9—12, IS, 14, 17—23, 26—28, 25, 29—30, 31, 32, 33—
111,115.

XL I— 30, 3i> 32, 34, 33, 35— 39.
XIL I— 5, 7, 9, 8, 10, 12, II, IS, 16—18.
XIIL 1—3, s, 4, 6, 7, 12, 9, 10, 8, II, 13, IS, 14, 16—18.

> See Klamroth, pp. 975—6, 38o» 383—4, 314— > 5* 3^6 ; Hdbeig, toL v. pp. xcvi, xcvii.

k

CH.VII] EUCLID IN ARABIA 8i

2. Definitions,

The Arabic omits the following definitions: iv. Deff. 3 — 7, vil.
Def. 9 (or 10), XI. Deff. 5 — 7, 15, 17, 23, 25 — 28; but it has the
spurious definitions VI. Deff. 2, 5, and those of proportion and ordered
proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges
V. Deff. II, 12 and also vi. Deff. 3, 4.

The order of the definitions is also different in Book vil. where,
after Def 11, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23,
and in Book XI. where the order is i, 2, 3,4, 8. 10, 9, 13, 14, 16, 12, 21,
22, 18, 19, 20, II, 24.

3. Lemmas andporisms.

All are omitted in the Arabic except the porisms to vi. 8, VIII. 2,
X. 3 ; but there are slight additions here and there, not found in the
Greek, e.g. in vill. 14. 15 (in K).

4. Alternative proofs.

These are all omitted in the Arabic, except that in X. 105, 106 they
are substituted for the genuine proofs; but one or two alternative
proofs are peculiar to the Arabic (Vl. 32 and vili. 4, 6).

The analyses and syntheses to xill. i — 5 are also omitted in the
Arabic.

Klamroth is inclined, on a consideration of all these differences, to
give preference to the Arabian tradition over the Greek (i) "on
historical grounds," subject to the proviso that no Greek MS. as
ancient as the 8th c. is found to contradict his conclusions, which are
based generally (2) on the improbability that the Arabs would have
omitted so much if they had found it in their Greek MSS., it being clear
from the Fihrist that the Arabs had already shown an anxiety for a
pure text, and that the old translators were subjected in this matter to
the check of public criticism. Against the " historical grounds," Heiberg
is able to bring a considerable amount of evidenced First of all there
is the British Museum palimpsest (L) of the 7th or the b^inning of
the 8th c. This has fragments of propositions in Book x. which are
omitted in the Arabic; the numbering of one proposition, which agrees
with the numbering in other Greek MS., is not comprehensible on
the assumption that eight preceding propositions were omitted in it,
as they are in the Arabic ; and lastly, the readings in L are tolerably
like those of our MSS., and surprisingly like those of B. It is also to
be noted that, although P dates from the loth c. only, it contains,
according to all appearance, an ante-Theonine recension.

Moreover there is positive evidence against certain omissions by
the Arabians. At-Tusi omits VI. 12, but it is scarcely possible that,
if Eutocius had not had it, he would have quoted VI. 23 by that
number*. This quotation of VI. 23 by Eutocius also tells against
Ishaq who has the proposition as VI. 25. Again, Simplicius quotes VI.
10 by that number, whereas it is VI. 13 in Ishaq ; and Pappus quotes,
by number, XIII. 2 (Ishaq 3, 4), XIII. 4 (Ishaq 8), XlII. 16 (Ishaq 19).

1 Heiberg in Zeitschrift fur Math, u, PhysiJk, XXIX., hist.-litt Abth. p. asqq.
' Apollonius, ed. Heibeig, vol. 11. p. 318, 3 — 5.

H. E. 6

it INTRODUCTION

On the other hand the contraction of IIL n, 12 into one propositton
in the Arabic tells in favour of the Arabic.

Further, the omission of certain porisms in the Arabic cannot be
supported; for Pappus quotes the porism to Xlll. 17*, P rod us those
to I [. 4, III, I, VI L 2\ and Simplicius that to iv, 15.

Lastly, some propositions omitted in the Arabic are required in
later propositions. Thus X. 13 is used in X, 18, 22, 23, 26 etc. ; X. 17
is wanted in x, 18, 26^ 36; Xlh 6, 13 are required for XIL 11 and XIL
15 respectively.

It must also be remembered that some of the things which were
properly omitted by the Arabians are omitted or marked as doubtful
in Greek MSS, also, especially in P, and others are rightly suspected for
other reasons (ag. a number of alternative proofs, lenr^mas, and porisms,
as well as the analyses and syntheses of Xlil. i — 5). On the other
hand, the Arabic has certain interpolations peculiar to our inferior
MSS, (cf the definition VL Def 2 and those of fitafiorlian and ordered
proportion).

Heiberg comes to the general conclusion that, not only is the
Arabic tradition not to be preferred offhand to that of the Greek MSs.,
but it must be regarded as inferior in authority. It is a question
how far the differences shown in the Arabic are due to the use of
Greek MSS, differing from those which have been most used as the
basis of our text, and how far to the arbitrary changes made by
the Arabians themselves. Changes of order and arbitrary omissions
could not surprise us, in view of the preface above quoted from the
Oxford MS, of Thabit-Ishaq, with its allusion to the many important
and necessary things left out by Abu *1 Wafa and to the author's
own rearrangement of Books XIL, XIII. But there is evidence of
differences due to the use by the Arabs of other Greek MSS. Heiberg'
is able to show considerable resemblances between the Arabic text
and the Bologna MS, b in that part of the MS. where it diverges so
remarkably from our other MSS. (see the short description of it above,
p, 49); in illustration he gives a comparison of the proofs of xtt, 7 in b
and in the Arabic respectively, and points to the omission in both of
the proposition given in Gregozy's edition as XI, 3S, and to a remark-
able agreement t>etween them as r^ards the order of the propositions
of Book XIL As above stated, the remarkable divergence of b only
affects Books xi. (at end) and xii. ; and Book xiii. in b shows none
of the transpositions and other peculiarities of the Arabic There
are many differences between b and the Arabic^ especially in the
definitions of Book XI., as well as in Book XIII. It is therefore a
question whether the Arabians made arbitrary changes, or the Arabic
form is the more ancient, and b has been altered through contact
with other MSS. Heiberg points out that the Arabians must be sdone
responsible for tlieir definition of a prism, which only covers a prism
with a triangular base. This could not have been Euclid's own, for
the word prism already has the wider meaning in Archimedes, and

* Pappus, V. p. 436, 5. • Prodiu, pp. 303 — ^4.

* ZeUsckr^fiir Maih.m. Pkysik^ XXIX., bist.-litt. Abth. p. 6aqq.

,

i

IF

'

CH. vii] EUCLID IN ARABIA 83

Euclid himself speaks of prisms with parallelograms and polygons
as bases (xi. 39 ; XII. 10). Moreover, a Greek would not have been
likely to leave out the definitions of the " Platonic " regular solids.

Heiberg considers that the Arabian translator had before him
a MS. which was related to b, but diverged still further from the rest
of our MSS. He does not think that there is evidence of the existence of
a redaction of Books I. — x. similar to that of Books XL, XII. in b ; for
Klamroth observes that it is the Books on solid geometry (xi. — XIII.)
which are more remarkable than the others for omissions and shorter
proofs, and it is a noteworthy coincidence that it is just in these
Books that we have a divergent text in b.

An advantage in the Arabic version is the omission of VII. DeC 10,
although, as lamblichus had it, it may have been deliberately omitted
by the Arabic translator. Another advantage is the omission of the
analyses and syntheses of xill. i — 5 ; but again these may have been
omitted purposely, as were evidently a number of porisms which
are really necessary.

One or two remarks may be added about the Arabic versions
as compared with one another. Al-Hajjaj*s object seems to have
been less to give a faithful reflection of the original than to write
a useful and convenient mathematical text-book. One characteristic
of it is the careful references to earlier propositions when their results
are used. ' Such specific quotations of earlier propositions are rare in
Euclid ; but in al-Hajjaj we find not only such phrases as "by prop,
so and so," " which was proved " or " which we showed how to do in
prop, so and so," but also still longer phrases. Sometimes he repeats
a construction, as in 1. 44 where, instead of constructing "the parallelo-
gram BEFG equal to the triangle C in the angle EBG which is equal
to the angle Z>" and placing it in a certain position, he produces AB
to G, making BG equal to half DE (the base of the triangle CDE in
his figure), and on GB so constructs the parallelogram BHKG by
I. 42 that it is equal to the triangle CDE, and its angle GBH is equal
to the given angle.

Secondly, al-Hajjaj, in the arithmetical books, in the theory of
proportion, in the applications of the Pythagorean 1. 47, and generally
where possible, illustrates the proofs by numerical examples. It is
true, observes Klamroth, that these examples are not apparently
separated from the commentary of an-NairIzi, and might not there-
fore have been due to al-Hajjaj himself; but the marginal notes to
the Hebrew translation in Munich MS. 36 show that these additions
were in the copy of al-Hajjaj used by the translator, for they expressly
give these proofs in numbers as variants taken from al-Hajjaj ^

These characteristics, together with al-Hajjaj*s freer formulation
of the propositions and expansion of the proofs, constitute an in-
telligible reason why Ishaq should have undertaken a fresh translation
from the Greek. Klamroth calls Ishaq's version a model of a good
translation of a mathematical text ; the introductory and transitional

1 Kkmroth, p. 310; Steinschneider, pp. 85 — 6.

6 — 2

84 INTRODUCTION *^ [ch. vu

phrases are stereotyped and few in number^ the technical terms are
simply and consistently rendered, and the less formal expressions
connect themselves as closely with the Greek as is consistent with
intelligibility and the character of the Arabic language. Only in
isolat^ cases does the formulation of definitions and enunciations
differ to any considerable extent from the original. In general, his
object seems to have been to get rid of difficulties and unevennesses
in the Greek text by neat devices, while at the same time giving a
faithful reproduction of it^

There are curious points of contact between the versions of
al-Iiajjaj and Thabit-Ishaq. For example, the definitions and
enunciations of propositions are often word for word the same»
Presumably this is owing to the fact that Ishaq found these de-
finitions and enunciations already established in the schools in his
time, where they would no doubt be learnt by heart, and refrained
from translating them afresh, merely adopting the older version with
some changes*. Secondly, there is remarkable agreement between
the Arabic versions as regards the figures, which show considerable
variations from the figures of the Greek text, especially as regards
the letters ; this is also probably to be explained in the same way,
all the later translators having most likely borrowed al^Hajjaj's
adaptation of the Greek figures'. Lastly, it is remarkable that the
version of Books XL — xni, in the Kjjrfbenhavn Ms. (K), purporting
to be by al-Hajjaj, is almost exactly the same as the Thabtt- Ishaq
version of the same Books in O. Klamroth conjectures that Ishaq
may not have translated the Books on solid geometry at all, and that
Thabit took them from al-yajjaj, only making some changes in order
to fit them to the translation of Ishaq*.

From the facts (i) that at-Tusi's edition had the same number
of propositions (468) as al-Hajjaj's version, while Thabit- Ishaq's had
478, and (2) that at-Tusi has the same careful references to earlier
propositions, Klamroth concludes that at-TusI deliberately preferred
al-^ajjaj's version to that of Ishaq*. Heiberg, however, points out
(i) that at-TusI left out Vl iz which, if we may judge by Klamroth's
silence, al-I;^ajjaj had, and (2) al-Hajjaj's version had one proposition
less in Books L and IlL than at-Tusi has. Besides^ in a passage quoted
by I:Iaji Khalfa* from at-TusI, the latter says that ** he separated the |
things which, in the approved editions, were taken from the archetype j
from the things which had been added thereto," indicating that he ;
had compiled his edition from dotA the earlier translations'. i

There were a large number of Arabian commentaries on, or ;
reproductions of, the Elements or portions thereof, which will be 1

1 Klamroth, p. 390, illnstrates Is^*s method hy hit way of distiiigaifhin^ i^aptnb^ i

(to be oongnient with) and 4^apiUfwBai fto be applied to), the confusion of which by timnt- J

laton was animadverted on by Savile. uStjIkxi avoided the confusion by using two entirely <

different words. ^

' Klamroth, pp. 310 — i. * ibid, p. aSy. '

^ iHd, pp. 304 — 5. * Uid. p. 374.

• tMd, pp. 304—5.

• miKhia&,i. p. 383.
' Hoberg, Zeitsckrififir

Math. m. Pkysik^ xxix., hist-litt. Abth. pp. a, 3.

CH. vii] EUCLID IN ARABIA 85

found fully noticed by Steinschneider^ I shall mention here the
commentators etc. referred to in the Fihrist, with a few others.

1. Abu '1 'Abbas al-Fadl b. Hatim an-NairlzI (bom at Nairiz,
died about 922) has already been mentioned*. His commentary
survives, as r^^rds Books I. — vi., in the Codex Leidensis 399, i, now in
course of publication by Besthorn and Heiberg, and as regards
Books I. — ^X. in the Latin translation made by Gherard of Cremona
in the nth c. and now published by Curtze from a Cracow MS.' Its
importance lies mainly in the quotations from Heron and Simplicius.

2. Ahmad b. 'Umar al-Kar&blsI (date uncertain, probably 9th —
loth c), ''who was among the most distinguished geometers and
arithmeticians*."

3. A1-' Abbas b. Sa'ld al-Jauharl (fl. 830) was one of the astro-
nomical observers under al-Ma*mun, but devoted himself mostly to
geometry. He wrote a commentary to the whole of the Elements^
from the beginning to the end ; also the " Book of the propositions
which he added to the first book of Euclid*."

4. Muh. b. 'Isa Abu 'Abdallah al-Mahtal (d. between 874 and
884) wrote, according to the Fihrist, (i) a commentary on Eucl.
Book v., (2) " On proportion," (3) " On the 26 propositions of the
first Book of Euclid which are proved without reductio ad absurdum^'
The work " On proportion " survives and is probably identical with, or
part of, the commentary on Book v.' He also wrote, what is not
mentioned by the Fihrht^ a commentary on Eucl. Book X., a fragment
of which survives in a Paris MS.*

5. Abu Ja'far al-Khftzin (i.e. " the treasurer " or '* librarian "), one
of the first mathematicians and astronomers of his time, was born in
Khurasan and died between the years 961 and 971. The Fikrist
speaks of him as having written a commentary on the whole of the
Elements*^ but only the commentary on the beginning of Book x.
survives (in Leiden, Berlin and Paris) ; therefore either the notes on
the rest of the Books have perished, or the Fikrist is in error *•. The
latter would seem more probable, for, at the end of his commentary,
al-Khazin remarks that the rest had already been commented on by
Sulaiman b. 'Usma (Leiden MS.)" or 'Oqba (Suter), to be mentioned
below. Al-Khazin's method is criticised unfavourably in the preface
to the Oxford MS. quoted by Nicoll (see p. y7 above).

6. Abii '1 Wafa al-BuzjanI (940-997), one of the greatest
Arabian mathematicians, wrote a commentary on the Elements, but

^ Steinschneider, Zeitschriftfur Math, u, Physik^ xxxi., hist.-litt. Abth. pp. 86 sqq.
' Steinschneider, p. 86, Pihrist (tr. Suter), pp. 16, 67 ; Suter, Die Mathematiier und
AHronomen tier Araber (1900), p. 45.

' Supplementum ad Eucltdis opera omnia^ ed. Heiberg and Menge, Leipzig, 1899.

* Fihrist^ pp. 16, 38 ; Steinschneider, p. 87 ; Suter, p. 65.

* Fihrisi^ pp. 16, 35; Steinschneider, p. 88; Suter, p. n.

* Fikrist, pp. 16, 25, 58.

' Suter, p. 36, note, quotes the Paris MS. 3467, 16** containing the work "on proportion'*
as the authority for this conjecture.

* MS. 3457, 39<» (cf. Woepcke in Mhn, pris, h Facad. des sciinces, xiv., 1856, p. 669).

* Fikrist, p. 17. *• Suter, p. 58, note b. " Steinschneider, p. 89.

86 INTKODUCnOH

^micu* Vtt I ^!

did not complete it*. His method is also unfavourably iri^rded in I
the same preface to the Oxford MS. 28a According to Haji Khalfa, he
also wrote a book on geometrical constructions, in thirteen chapters.
Apparently a book answering to this description was compiled by a
gifted pupil from lectures by Abu 1 Wafa, and a Paris MS. (Anc fends
169) contains a Persian translation of this work^ not that of Abu '1 Wafa
himself. An analysis of the work was given by Wocpcke', and some
particulars will be found in Cantor'* Abu '1 Wafa also wrote a
commentary on Diophantus, as well as a separate " book of proofs
to the propositions which Diophantus used in his book and to what
he (Abu '1 Wafa) employed in his commentary*/'

7. Ibn Rfthawaihl al-ArjinT also commented on EucI, Book X,*

8. 'All b. Ahmad Abu U-Qasim al-AntikI (d. 987) wrote a
commentary on the whole book*; part of it seems to survive (from
the 5th Book onwards) at Oxford (CataL MSS. orient. II. 281 )^

9. Sind b. *Ali Abu *t-Taiyib was a Jew who went over to
Islam in the time of aKMa'niun, and was received among his astro-
nomical observers, whose head he became" (about 830) ; he died after
864. He wrote a commentary on the whole of the EkmtHts ; " Abu
*AlI saw nine books of it^ and a part of the tenthV His book " On
the Apotomae and the Medials " mentioned by the Fikrisi^ may be
the same as, or part of, his commentary on Book X*

10. Abu Yusuf Ya'qub b. Muh. ar-Razf '^ wrote a commentafy
on Book X., and that an excellent one, at the instance of Ibn al-
*AmId"."

11. The Fihrisi next mentions al> Kindt (Abu Yusuf Ya'qQb b.
Ishaq b. as-Sabbah al-Kindi, d. about 873), as the author (1) of a
work " on the objects of Euclid's book/' in which occurs the statement
that the Elements were originally written by Apollonius, the carpenter
(see above, p. 5 and note), (2) of a book ''on the improvement of
Euclid's work," and (3) of another " on the improvement of the 14th
and 15th Books of Euclid." ''He was the most distinguished man
of his time, and stood alone in the knowledge of the old sciences
collectively ; he was called ' the philosopher of the Arabians ' ; his
writings treat of the most different branches of knowledge, as logic,
philosophy, geometry, calculation, arithmetic, music, astronomy and
others"." Among the other geometrical works of al-Kindi mentioned \
by the Fihrisi^ are treatises on the closer investigation of the results !
of Archimedes concerning the measure of the diameter of a circle in ^'
terms of its circumference, on the construction of the figure of the two \
mean proportionals, on the approximate determination of the chords 1

• Fikristt p. 17. I

• WoepdLcyimmaiAnaH^tte, Sit. v. T. v. pp. «i8— 956 and 300—350. l

• Geuk. d. Math, vol. ig, pp. 743—6.
« mhriti^ p. 39 ; Suter, p. 71. • Rkrisi, p. 17 ; Sater, p. 17.

• Rkrisi^ p. 17. f Suter, p. 64.

• Fihrisi^ p. 17, 39 ; Suter, pp. 13, 14. • Hhri$t, p. 17.
M Fihrist^ p. 17 ; Suter, p. 66. » FihrisU p. 17, ia-15.
^ The mere catalogue of al-Kindl't works on the various boAclies of idence takes up

Ibpr octavo pages (i i— 15) of Suter's translation of the Ftkrist.

1

CH. vii] EUCLID IN ARABIA 87

of the circle, on the approximate determination of the chord (side) of
the nonagon, on the division of triangles and quadrilaterals and con-
structions for that purpose, on the manner of construction of a circle
which is equal to the surface of a given cylinder, on the division of
the circle, in three chapters etc.

12. The physician Nazlf b. Yumn (or Yaman) al-Qass ("the
priest") is mentioned by the Fihrist as having seen a Greek copy
of Eucl. Book X. which had 40 more propositions than that which
was in general circulation (containing 109), and having determined
to translate it into Arabic^ Fragments of such a translation exist
at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab, in
Woepcke's tract); No. 18 contains "additions to some propositions
of the loth Book, existing in the Greek language"." Nazlf must have
died about 990'.

13. Yu^iannft b. Yusuf b. al-Iiarith b. al-Bitnq al-Qass (d. about
980) lectured on the Elements and other geometrical books, made
translations from the Greek, and wrote a tract on the " proof" of the
case of two straight lines both meeting a third and making with it,
on one side, two angles together less than two right angles*. Nothing
of his appears to survive, except that a tract "on rational and irrational

^ magnitudes," No. 48 in the Paris MS. just mentioned, is attributed
' to hinL

I 14. Abu Muh. al-Hasan b. 'Ubaidallah b. Sulaiman b. Wahb

. (d. 901) was a geometer of distinction, who wrote works under the
y two distinct titles " A commentary on the difficult parts of the work
r of Euclid " and " The Book on Proportion*." Suter thinks that an-
other reading is possible in the case of the second title, and that it
may refer to the Euclidean work "on the divisions (of figures)*."
, 15. Qustft b. Luq§ al-Ba'labakki (d. about 912), a physician,

philosopher, astronomer, mathematician and translator, wrote " on the
difficult passages of Euclid's book" and "on the solution of arith-
metical problems from the third book of Euclid'"; also an "intro-
duction to geometry," in the form of question and answer".

16. Thabit b. Qurra (826-901), besides translating some parts
of Archimedes and Books V. — vii. of the Conies of Apollonius, and
revising Ishaq*s translation of Euclid's Elements^ also revised the trans-
lation of the Data by the same Ishaq and the book On divisions of
figures translated by an anonymous writer. We are told also
that he wrote the following works: (i) On the Premisses (Axioms,
Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On
the propositions and questions which arise when two straight lines
are cut by a third (or on the " proof" of Euclid's famous postulate).
The last tract is extant in the MS. discovered by Woepcke (Paris
2457, 32®). He is also credited with "an excellent work" in the
shape of an " Introduction to the Book of Euclid," a treatise on

* Fikristf pp. 16, 17.

" Woepcke, Mhn.pris. h Vacad, des sdenut^ XI v. pp. 666, 668.

» Suter, p. 68. * Fihnst, p. 38 ; Suter, p.

• Fihrist^ p. «6, and Suter's note, p. 60. • Suter, p. an, note 13.

' Fihrist, p. 43. • Fihrist^ p. 43 ; Suter, p.

60.
41.

wm

88 INTRODUCTION [ch. vii

I

Geometry dedicated to fsma^n b. Bulbul, a Compendium of Geometry,
and a large number of other works for the titles of which reference
may be made to Sutcr, who also gives particulars as to which are
extant^

17. Abu Sa'ld Sinin b* Thabit b, Qurra, the son of the translator
of Euclid, followed in his father's footsteps as geometer, astronomer
and physician. He wrote an ''improvement of the book of ,.-,.. on
the Elements of Geometry, in which he made various additions to the
original" It is natural to conjecture that Emlid is the name missing
in 3iis description (by Ibn abl Usaibi'a); Casiri has the name Aqaton*.
The latest eaitor of the Tdrlkh al-Hukantd^ however^ makes the name
to be Iflaton (« Plato), and he refers to the statement by the Fihrist
and Ibn" al-QiftI attributing to Plato a work on the Elements of
Geometry translated by Qusta. It is just possible, therefore, that at
the time of Qusta the Arabs were acquainted with a book on the
Elements of Geometry translated from the Greek, which they attri-
buted to Plato*. Sinan died in 943.

18. Abu Sahl Wijan (or Waijan) b. Rustam al-Kuhl (fl, 988),
bom at Kuh in Tabaristan, a distinguished geometer and astronomer,
wrote, according to the Fihrist, a " Book of the Elements " after that
of Euclid^; the ist and 2nd Books survive at Cairo, and a part of
the 3rd Book at Berlin {^<^22Y, He wrote also a number of other
geometrical works: Additions to the 2nd Book of Archimedes on .
the Sphere and Cylinder (extant at Paris, at Leiden* and in the India (
Office), On the finding of the side of a heptagon in a circle (India
Office and Cairo), On two mean proportionals (India OfficeX which
last may be only a part of the Additions to Archimedes' On the Sphere -%
and Cylinder, etc '\

19. Abu Nasr Muh. b. Muh. b. Tarkhan b. Uzlag al-FtrftbT
(870-950) wrote a commentary on the difficulties of the introductory
matter to Books I. and v.* This appears to survive in the Hebrew
translation which is, with probability, attributed to Moses b. Tibbon^ i

2a Abu 'All al-^asan b. al-Hasan b. al-Haitham (about 965- ]
1039), known by the name Ibn al-Haitham or Abu 'All al-Basrl, was a
man of great powers and knowledge, and no one of his time approached I
him in the field of mathematical science. He wrote several works on
Euclid the titles of which, as translated by Woepcke from Usaibi'a, i
are as follows"

1. Commentary and abridgment of the Elements. i

2. Collection of the Elements of Geometry and Arithmetic, f
drawn from the treatises of Euclid and ApoUonius. j

3. Collection of the Elements of the Calculus deduced from
the principles laid down by Euclid in his Elements. \

* Sater, pp. 54—8.

* Purist (ed. Suter), p. 59, note 13a ; Suter, p. 51, note b.

■ See Suter in Bibhatkeca Matkematica, iv„ 1903-4, pp. 196—7, reriew of Julias
Lippert's Km ai-Qifii. Ta'ruh al-kukamd, Leiprig, 1903.

* Fikristt p. 40. • Suter, p. 75.

* Suter, p. 55. f SteioKhneider, p. 9«.

* Stcintcnncider, pp. 93 — ^3.

I

r

i

1

CH. vii] EUCLID IN ARABIA 89

4. Treatise on "measure" after the manner of Euclid's
Elements.

5. Memoir on the solution of the difficulties in Book i.

6. Memoir for the solution of a doubt about Euclid, relative
to Book V.

7. Memoir on the solution of a doubt about the stereometric
portion.

8. Memoir on the solution of a doubt about Book Xil.

9. Memoir on the division of the two magnitudes mentioned
in X. I (the theorem of exhaustion).

ID. Commentary on the definitions in the work of Euclid
(where Steinschneider thinks that some more general expression
should be substituted for " definitions ").

The last-named work (which Suter calls a commentary on the
Postulates of Euclid) survives in an Oxford MS. (CataL MSS. orient.
I. 908) and in Algiers (1446, i®).

A Leiden MS. (966) contains his Commentary "on the difficult
places " up to Book v. We do not know whether in this commentary,
which the author intended to form, with the commentary on the
Musadarat, a sort of complete commentary, he had collected the
separate memoirs on certain doubts and difficult passages mentioned
in the above list.

A commentary on Book v. and following Books found in a
Bodleian MS. (Catal. II. p. 262) with the title " Commentary on Euclid
and solution of his difficulties " is attributed to b. Haitham ; this might
be a continuation of the Leiden MS.

The memoir on x. i appears to survive at St Petersburg, Ma de
rinstitut des langues orient 192, 5* (Rosen, Catal. p. 125).

21. Ibn SInft, known as Avicenna (980-1037), wrote a Com-
pendium of Euclid, preserved in a Leiden MS. No. 1445, and forming
the geometrical portion of an encyclopaedic work embracing Logic,
Mathematics, Physics and Metaphysics^

22. Ahmad b. al-Husain al-AhwftzT al-Katib wrote a com-
mentary on Book X., a fragment of which (some 10 pages) is to be
found at Leiden (970), Berlin (5923) and Paris (2467, i8'*)*.

23. Naslraddin at-Tusi (i 201-1274) who, as we have seen,
brought out a Euclid in two forms, wrote:

1. A treatise on the postulates of Euclid (Paris, 2467, 5®).

2. A treatise on the 5th postulate, perhaps only a part of
the foregoing (Berlin, 5942, Paris, 2467, 6®).

3. Principles of Geometry taken from Euclid, perhaps
identical with No. i above (Florence, Pal. 298).

4. 105 problems out of the Elements (Cairo). He also edited
the Data (Berlin, Florence, Oxford etc.)'.

24. Muh. b. Ashraf Shamsaddin as-Samarqandl (fl. 1276) wrote
" Fundamental Propositions, being elucidations of 35 selected proposi-

^ Steinschneider, p. 91 ; Suter, p. 89. ' Suter, p. 57.

• Suter, pp. 150 — I.

u

^a

90 INTRODUCTION [cm. vii

tions of the first Books of Euclid " which are extant at Gotha (1496
and 1497), Oxford (Catal l. 967, 2^), ^^^ Brit Uus.\

25, Musa b- Muh. b* Mahmud, known as Qa^izade ar-Ruml (i,c»
the son of the judge from Asia Minor), who died between 1436 and
1446, wrote a commentary on the "Fundamental Propositions" ju^t
mentioned^ of which many MSS* are extant'. It contained biographical
statements about Euclid alluded to above (p. S, note),

26, Abu Da'ud Sulaiman b, 'Uqba^ a contemporary of aKKhazin
(see above, No* 5), wrote a commentary on the second half of Book x.,
which is, at least partly, extant at Leiden (974) under the title " On
the binomials and apotomae found in the lOth Book of Euclid V

27, The Codex Letdensis 399^ i containing al-Hajjaj*s transla-
tion of Books L — VL is said to contain glosses to it by Sa'id b* Mas'ud
b. al-Qass, apparently identical with Abu Nasr Gars al-Na'ma, son of
the physician Mas'^Od b. al-Qass al-Bagdadi, who lived in the time of
the last Caliph al-Musta'sini (d. 1258)*,

28, Abu Muhammad b. Abdalbiql al-Bagdadt al*Fara^i (d*
1 141, at the age of over 70 years) is stated in the Tdrtkh al-Hukamd
to have written an excellent commentary on Book x. of the Elements,
in which he gave numerical examples of the propositionsV This is
published in Curtze's edition of an-Nainzi where it occupies pages
252 — 386*

29, Yahya b, Muh. b, 'Abdan b. *Abdalwahid, known by the
name of Ibn al-Lubudl (1210-126S), wrote a Compendium of Euclid^
and a short presentation of the postulates'.

30, Abu 'Abdallah Muh. b, Mu'adh al-Jayyanl wrote a com-
mentary on Eucl, Book v. which survives at Algiers (1446, 3*)'.

31, Abu Nasr Mansur b* 'All b. 'Iraq wrote^ at the instance of
Muh. b, Ahmad Abu V-Raihan al-Blruni (973-1048), a tract "on
a doubtful (difficult) passage in Eucl Book XIII/' (Berlin, S925)'.

' Stit«T, p. 157- * *^*^' P" *75- * *^" P- 5**'

< ihid. pp, [55— 4 i *V'

° GaiU, i^ i4 ; Suloschiieider, pp. 94 — 5.

* Safer in BWioiMsca MaiAemaiua^ iv^, 1904, pp. «5» 195 ; Sater has also an artide on
its contents, BiUioikica MaiJkimaiuat viu, 1906^71 pp. 354 — 351.

^ Steinschneider, p. 94 ; Sater, p. 14O.

> Sater, Nacktrdge und Berickttgwtgen^ in Abkandimngen mr Gesck. der math, fVUsin-
sehapen^ xiv., 1903, p. 17a

* Snter, p. 81, ana Na€htrag$^ p. 173.

CHAPTER VIII.

PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS.

Cicero is the first Latin author to mention Euclid^; but it is not
likely that in Cicero's time Euclid had been translated into Latin or
was studied to any considerable extent by the Romans; for, as Cicero
says in another place', while geometry was held in high honour
among the Greeks, so that nothing was more brilliant than their
mathematicians, the Romans limited its scope by having regard only
to its utility for measurements and calculations. How very little
theoretical geometry satisfied the Roman agrimensares is evidenced
by the work of Balbus de mensuris^^ where some of the definitions of
Eucl. Book I. are given. Again, the extracts from the Elements found

. in the fragment attributed to Censorinus (fl. 238 A.D )* are confined to

I the definitions, postulates, and common notions. But by degrees the
Elements passed even among the Romans into the curriculum of a

! liberal education ; for Martianus Capella speaks of the effect of the
enunciation of the proposition "how to construct an equilateral
triangle on a given straight line " among a company of philosophers,

' who, recognising the first proposition of the Elements, straightway
break out into encomiums on Euclid*. But the Elements were then

; {c. 470 A.D.) doubtless read in Greek ; for what Martianus Capella
gives* was drawn from a Greek source, as is shown by the occurrence
of Greek words and by the wrong translation of I. def. i (" punctum
vero est cuius pars nihil est"). Martianus may, it is true, have
quoted, not from Euclid himself, but from Heron or some other ancient
source.

But it is clear from a certain palimpsest at Verona that some
scholar had already attempted to translate the Elements into Latin.
This palimpsest' has part of the " Moral reflections on the Book of
Job " by Pope Gregory the Great written in a hand of the 9th c. above
certain fragments which in the opinion of the best judges date from
the 4th c Among these are fragments of Vergil and of Livy, as well
as a geometrical fragment which purports to be taken from the 14th
and 15th Books of Euclid. As a matter of fact it is from Books XIL
and XIII. and is of the nature of a free rendering, or rather a new

* Dearatort III. 13a. • Tusc, I. 5.

* GromaHH veteres, I. 97 sq. (ed. F. Blume, K. Lachmann and A. Rudorff. Berlin,
1848, 1851).

* Censorinus, ed. Hultsch, pp. 60— 3.

* Martianus Capella, vi. 734. * ibid, vi. 708 sq.
^ Cf. Cantor, i„ p. 565.

:;

INTRODUCTION [ch, viii

arrangement, of Euclid with the propositions in different orders The
MS- was evidently the translator's own copy, because some words arc
struck out and replaced by synonyms. We do not know whether the
translator completed the translation of the whole* or in what relation
his version stood to our other sources.

Magnus Aurelius Cassiodorius (b* about 475 a.d.) in the geometrical
part of his encyclopaedia Deartibus ac disciplinis Hberalium literarum
says that geometry was represented among the Greeks by Euclid,
Apollonius, Archimedes, and others, "of whom Euclid was given us
translated into the Latin language by the same great man Boethius**;
also in his collection of letters* is a letter from Theodoric to Boethius
containing the words, "for in your translations ... Ntcomachus the
arithmetician, and Euclid the geometer^ are heard in the Ausonian
tongue" The so-called Geometry of Boethius which has come down
to us by no means constitutes a translation of Euclid The MSS.
variously give five, four, three or two Books, but they represent only
two distinct compilations, one normally in five Books and the other
in two. Even the latter, which was edited by Friedlein, is not
genuine*, but appears to have been put together in the nth c», from
various sources. It begins with the definitions of EucL I,, and in these
are traces of perfectly correct readings which are not found even in
the MSS. of the loth c, but which can be traced in Proclus and other
ancient sources ; then come the Postulates (five only), the Axioms .
(three only), and after these some definitions of Eucl 11., 11 L, iv.
Next come the enunciations of Eucl I., of ten propositions of Book 11,,
and of some from Books llL, IV., but always without proofs ; there
follows an extraordinary passage which indicates that the author will
now give something of his own in elucidation of Euclid, though what '
follows is a literal translation of the proofs of EucL I. I — 3, This
latter passage, although it affords a strong argument against the
genuineness of this part of the work, shows that the Pseudoboethius
had a Latin translation of Euclid from which he extracted the three
propositions.

Curtze has reproduced, in the preface to his edition of the trans-
lation by Gherard of Cremona of an-Nairizf s Arabic commentary on
Euclid, some interesting fragments of a translation of Euclid taken
from a Munich MS. of the loth c. They are on two leaves used
for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis
2*^ 757) atr>d consist of portions of Eucl, L 37, 38 and IL 8, translated
literally word for word from the Greek text The translator seems to
have been an Italian (c£ the words '' capitolo nono " used for the ninth
prop, of Book II.) who knew very little Greek and had moreover little
mathematical knowledge. For example, he translates the capital letters
denoting points in figures as if they were numerals : thus tA ABF,

* The frugment wms deciphered by W. Stndemmid, who oommimicated his resalts to
Cantor.

* CaiBodorius, Varioi^ i. 45, p. 40, 13 ed. Mommieii.

* See etpedaUy Weissenbom in AbkatuUumgm mr Getck, d. Math. II. p. 185 iq.;
1 Heibcsg in Pkilol^gus^ XLUI. p. 507 sq. ; Cantor, ig, p. 580 sq.

CH. viii] TRANSLATIONS AND EDITIONS 93

AEZ is translated ''que primo secundo et tertio quarto quinto et
scptimo," T becomes "tricentissimo " and so on. The Greek MS. which
he used was evidently written in uncials, for AEZ6 becomes in one
place " quod autem septimo nono," showing that he mistook AE for
the particle &', and koI 6 2TU is rendered ''sicut tricentissimo et
quadringentissimo/' showing that the letters must have been written
KAIOCTU.

The date of the Englishman Athelhard (iEthelhard) is approxi-
mately fixed by some remarks in his work Perdiffidles Quaestiones
Naturales which, on the ground of the personal allusions they contain,
must be assigned to the first thirty years of the 12th c* He wrote a
number of philosophical works. Little is known about his life. He
is said to have studied at Tours and Laon, and to have lectured at the
latter school. He travelled to Spain, Greece, Asia Minor and Egypt,
and acquired a knowledge of Arabic, which enabled him to translate
from the Arabic into Latin, among other works, the Elements of
Euclid. The date of this translation must be put at about 11 20.
MSB. purporting to contain Athelhard's version are extant in the
British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll.
47 and Ball. Coll. 257 of 12th c), Nurnberg (Johannes Regiomontanus'
copy) and Erfurt

Among the very numerous works of Gherard of Cremona (i 1 14 —
1 1 87) are mentioned translations of ** 15 Books of Euclid" and of the
Data}. Till recently this translation of the Elements was supposed to
be lost; but Axel Anthon Bjombo has succeeded (1904) in discovering
a translation from the Arabic which is different from the two others
known to us (those by Athelhard and Campanus respectively), and
which he, on grounds apparently convincing, holds to be Gherard's.
Already in 1901 Bjombo had found Books x. — xv. of this translation
in a MS. at Rome (Codex R^inensis lat. 1268 of 14th c.)'; but three
years later he had traced three MSB. containing the whole of the same
translation at Paris (Cod. Paris. 7216, 15th c), Boulogne-sur-Mer
(Cod. Bononiens. 196, 14th c), and Bruges (Cod. Brugens. 521, 14th c),
and another at Oxford (Cod. Digby 174, end of 12th c.) containing a
fragment, XI. 2 to XIV.* The occurrence of Greek words in tfiis
translation such as rombus^ romboides (where Athelhard keeps the
Arabic terms), ambligonitis, orthogonius, gnomo^ fyramis etc., show
that the translation is independent of Athelhard's. Gherard appears
to have had before him an old translation of Euclid from the Greek
which Athelhard also often followed, especially in his terminology,
using it however in a very different manner. Again, there are some
Arabic terms, e.g. meguar for axis of rotation^ which Athelhard did not
use, but which is found in almost all the translations that are with
certainty attributed to Gherard of Cremona; there occurs also the

» Cantor, Gesch, d. Math, i„ p. 006.

* Boncompagni, Delia vita e delU cfere di Gherardo Crewunuse^ Rome, 1851, p. 5.

* Described in an appendix to Studien iiber Menelaos^ Sphdrik {Abhanduungm %ur
Gtschuhte der mathematischen IVissenschafien, xiv., 1901).

* Sec Bibliotheca Mathematica, vig, 1905-^, pp. 141—8.

94 INTRODUCTION [ck. viii

expression "superficies equidjstantium latcnim et rectomm angulorum/'
found also in Gherard's translation of an-NairM, where Athelhard says
"parallelogrammum rectan^lum/' The translation is much clearer
than Athelhard's: it is neither abbreviated nor "edited'' as Athelhard's
appears to have been ; it is a word-for*word translation of an Arabic
MS. containing a revised and critical edition of Thabit^s version. It
contains several notes quoted from Thabit himself {Thebit dixii\ e.g.
about alternative proofs etc* which Thabit found ''in another Greek
MS./* and is therefore a further testimony to Thabit s critical treatment
of the text after Greek MSS. The new editor also added critical
remarks of his own* e.g. on other proofs which he found in other
Arabic versions, but not in the Greek; whence it is clear that he
compared the Thabit version before him with other versions as care-
fully as Thabit collated the Greek MSS. Lastly, the new editor speaks
of ''Thebit qui transtulit hunc librum in arabicam linguam" and of
"translatioThebit," which may tend to confirm the statement of al-Qifti
who credited Thabit with an independent translation, and not (as the
Fihrist does) with a mere improvement of the version of Ishaq b,
^unain.

Gherard's translation of the Arabic commentary of an-NatiizI on
the first ten Books of the EUments was discovered by Maximitian
Curtze in a MS. at Cracow and published as a supplementary volume
to Heiberg and Menge's Euclid^: it will often be referred to in this
work.

Next in chronological order comes Johannes Campanus (Cam pane)
of Novara. He is mentioned by Roger Bacon (12(4-1294) as a
prominent mathematician of his time*, and this indication of his date
is confirmed by the fact that he was chaplain to Pope Urban IV, who
was Pope from 1261 to 1281*. His most important achievement was
his edition of the Elements including the two Books XIV. and XV.
which are not Euclid's. The sources of Athelhard's and Campanus'
translations, and the relation between them, have been the subject of
much discussion, which does not seem to have led as yet to any
definite conclusion. Cantor (III, p. 91) gives references^ and some
particulars. It appears that there is a MS. at Munich (Cod. lat. Mon.
1 3021) written by Sigboto in the 12th c at Priifning near Regensburg,
and denpted by Curtze by the letter R, which contains the enunciations
of part of Euclid. The Munich MSS. of Athelhard and Campanus'
translations have many enunciations textually identical with those in
R, so that the source of all three must, for these enunciations, have

' AnariHi in decern HbrM friores EUmenUrum EucUdis Commentarii ex inter^niaUcmi
Gherardi Cremonensis in codice Cracaviensi 569 servata edidit Maximiliaous Cunze, Leiptig
(Teubner), 1899.

* Cantor, Hi, p. 88.

' Tiraboschi, Storia della leiteratura ita/iana, IV. 145—160.

* H. V^eissenbom in ZeUschrift fiir Math, «. Pkysik^ xxv., Supplement, pp. T43— 166,
and in his monograph, Die Obenettungen des Ettkliddurck Campano und ZambertiXi^i) ;
Max. Curtze in FhtloUgiiche Rundschau (1881), I. pp. 943^950, and in Jdhresbericht iiAtr
die Forischritte der classischen AUerihumswissemchaft^ XL. (1C64, in.) pp. 19 — 39 ; Heibeig
in ZaischriftJUr Math, u. Physik^ XXXV., hist.-litt Abth., pp. 48—58 and pp. 81—6.

CH. viii] TRANSLATIONS AND EDITIONS 95

been the same; in others Athelhard and Campanus diverge com-
pletely from R, which in these places follows the Greek text and is
therefore genuine and authoritative. In the 32nd definition occurs the
word " elinuam," the Arabic term for " rhombus," and throughout the
translation are a number of Arabic figures. But R was not translated
from the Arabic, as is shown by (among other things) its close
resemblance to the translation from Euclid given on pp. 377 sqq. of
the Gromatici Veteres and to the so-called geometry of Boethius. The
explanation of the Arabic figures and the word " elinuam " in Def. 32
appears to be that R was a late copy of an earlier original with
corruptions introduced in many places ; thus in Def. 32 a part of the
text was completely lost and was supplied by some intelligent copyist
who inserted the word "elinuam," which was known to him, and also
the Arabic figures. Thus Athelhard certainly was not the first to
translate Euclid into Latin ; there must have been in existence before
the nth c. a Latin translation which was the common source of R,
the passage in the Gromatici^ and '' Boethius." As in the two latter
there occur the proofs as well as the enunciations of L i — 3, it is
possible that this translation originally contained the proofs also.
Athelhard must have had before him this translation of the
enunciations, as well as the Arabic source from which he obtained his
j proofs. That some sort of translation, or at least fragments of one,
were available before Athelhard's time even in England is indicated
by some old English verses^ :

*'The clerk Euclide on this wyse hit fonde
Thys craft of gemetry yn Egypte londe
Yn Egypte he tawghte hyt ful wyde,
In dyvers londe on every syde.
Mony erys afterwarde y understonde
Yer that the craft com ynto thys londe.
Thys craft com into England, as y yow say,
Yn tyme of good kyng Adelstone's day,"

which would put the introduction of Euclid into England as far back
as 924-940 A.D.

We now come to the relation between Athelhard and Campanus.
That their translations were not independent, as Weissenborn would
have us believe, is clear from the fact that in all MSS. and editions,
apart from orthographical differences and such small differences as
are bound to arise when MSS. are copied by persons with some
knowledge of the subject-matter, the definitions, postulates, axioms,
and the 364 enunciations are word for word identical in Athelhard
and Campanus ; and this is the case not only where both have the
same text as R but where they diverge from it. Hence it would seem
that Campanus used Athelhard's translation and only developed the
proofs by means of another redaction of the Arabian Euclid. It is
true that the difference between the proofs of the propositions in the
two translations is considerable; Athelhard's are short and com-

^ Quoted by Halliwell in Rara Maihematica (p. 56 note) from MS. Bib. Reg. Mas. Brit.
17 A. I. f. 2^—3.

96 INTltOmTCTICni

pressed, Campanus' cleamr and mote c o mp lete, iolkMrlaff tile Gnei
text more closely, thoadi still at aome diftaiioe» Ftetibtt^: lb
arrangement in the two is diflerent; in AtbeUianl the praoft itgitM i ty | j
precede the enunciation^ Dunpanos §oXkmm tlie mual ctdm: ft iaa
question how far the differencet in the pnxA, and certain additiwiJtt
each, are due to the two transtatoia tfaantdves or go back to AmUc
originals. The latter supposition seems to Cocttt and Cantor the
more probable one. Curtae's general view of tiie relation of Qininan n s
to Athelhard is to the effect that Adidhard's tinnslalion was nwiMllj
altered, from the form in iHiidi it appears in the two Emmt Wm.
described by Weissenbom, fay socoesahre copyists and ciwiwi i mta tor s
wAo had Arabic originmU m^itm tlmm^ w^ it^ took ti» faraa irilidi
Campanus gave it and in which it was pnUished In soppoit of ttis
view Curtze refers to Rq;iomontaniis*copy of the AtfadharcMranipanns
translation. In R^omontanus' own prc&ce the title is given, 'Md
this attributes the translation to Atfidhard; bit^ wiiite diia copy
agrees almost exactly wiUi Athelhard In Book L, yet, in places wtoa
Campanus is more lengthy, it has similar additions, and in the Isler
Books, especially from Book Ul. onwards, agrees ahsdittd^ wfth
Campanus; R^iomontanu% too^ himself implies tinrt, fhrnMh A
translation was Athelhard's, Campanus had reidsed it; fcr ha !■
notes in the margin such as the following, **Campani est hec^" ''dnfako
an demonstret hie Campanus " etc. *

We come now to the printed editions of tlie udiole or of portfams
of the ElemenU, This is not the idace for a complete bihlkiipaplqr» :
such as Riccardi has attempted in his valuable memoir issued in five
parts between 1887 and 1893, which makes a large book in itself ^
I shall confine myself to saying something of the most noteworthy
translations and editions. It will be convenient to give first the Latin
translations which preceded the publication of the editio prinups of
the Greek text in 1 533, next the most important editions of the Greek
text itself, and after them the most important translations arranged
according to date of first appearance and languages, first the Latin
translations after 1533, then the Italian, German, French and English
translations in order.

It may be added here that the first allusion, in the West, to the
Greek text as still extant is found in Boccaccio's commentary on the
Divina Commedia of Dante*. Next Johannes Regiomontanus, who
intended to publish the Elements after the version of Campanus, but
with the latter^s mistakes corrected, saw in Italy (doubtless when
staying with his friend Bessarion) some Greek MSS. and noticed how
far they differed from the Latin version (see a letter of his written in |
the year 147 1 to Christian Roder of Hamburg)*.

^ Saggw di una Bibliogrt^ EucUdea^ memoria del Prof. Pietio Riccardi (Bologiui, !
1887, 1888, 1890, 1893). ^

" I. p. 404.

* Pabliahed in C. T. de Mail's fitmorabilia BibUatiucarum NorimUrgpumm^ Pttrt I. '
p. 190 iqq.

i

i CH. viii] TRANSLATIONS AND EDITIONS 97

I

I. Latin translations prior to 1533.

I 1482. In this year appeared the first printed edition of Euclid,

which was also the first printed mathematical book of any import-
ance. This was printed at Venice by Erhard Ratdolt and contained
Campanus' translation \ Ratdolt belonged to a family of artists at
Augsburg, where he was born about 1443. Having learnt the trade
of printing at home, he went in 1475 to Venice, and founded there a
famous printing house which he managed for 1 1 years, after which he
returned to Augsburg and continued to print important books until
1 5 16. He is said to have died in 1528. Kastner' gives a short
description of this first edition of Euclid and quotes the dedication to
Prince Mocenigo of Venice which occupies the page opposite to the
first page of text. The book has a margin of 2^ inches, and in this
margin are placed the figures of the propositions. Ratdolt says in
his dedication that at that time, although books by ancient and
modern authors were printed every day in Venice, little or nothing
mathematical had appeared : a fact which he puts down to the diffi-

Iculty involved by the figures, which no one had up to that time
succeeded in printing. He adds that after much labour he had
discovered a method by which figures could be produced as easily as
letters*. Experts are in doubt as to the nature of Ratdolt's discovery.
Was it a method of making figures up out of separate parts of figures,
straight or curved lines, put together as letters are put together to
make words > In a life of Joh. Gottlob Immanuel Breitkopf, a con-
temporary of Kastner's own, this member of the great house of
Breitkopt is credited with this particular discovery. Experts in that
^same house expressed the opinion that Ratdolt's figures were wood-
'cuts, while the letters denoting points in the figures were like the
other letters in the text ; yet it was with carved wooden blocks that
printing b^an. If Ratdolt was the first to print geometrical figures,
it was not long before an emulator arose ; for in the very same year

IMattheus Cordonis of Windischgratz employed woodcut mathematical
figures in printing Oresme*s De laHtudinibus\ How eagerly the
opportunity of spreading geometrical knowledge was seized upon is
proved by the number of editions which followed in the next few
years. Even the year 1482 saw two forms of the book, though they
only differ in the first sheet Another edition came out in i486
{Ulmae, apud lo, Regerutn) and another in 149 1 {Vincentiae per

^ Curtze (An-NairizI, p. xiii) reproduces the heading of the first page of the text as

follows (there is no title-pa^e) : PreclarifTimu opus elemento^ Eucliois megarefis vna cu

cOmentis Camponi pfpicacifnini in arte geometric incipit felicit*, after which the definitions

begin at once. Other copies have the shorter heading : Preclarissimus liber elementonim

Euclidis perspicacissimi : in artem Geometric incipit quam foelicissime. At the end stands

I the following : C Opus elementoru euclidis megarenf is in geometria arte Jn id quoq} Campani

I pfpicaciffinu C6mentationes finiut. Erhardus ratdolt Augustensis imprefTor folertiffimus .

; venetijs iropreirit . Anno ialutis . M.cccc.lxxxij . Octauis . Calefi . Jufi . Lector . Vale.

t' Kastner, GtschUhte dtr Maihematik^ I. p. 189 sqq. See also Weissenbom, Die Obersett-
ungen des EukHd durch Campano und Zamoertit pp. i — 7.
' "Mea industria non sine maximo labore efieci vt qua fturilitate litterarum elementa
imprimuntur ea etiam geometrice figure conficerentur."

* Curtze in Zeiischriftfiir Math, u, Physik, XX., hist.-litt. AbUi. p. 58.

H. E. 7

I

r

98 INTRODUCTION [cH.Tin

Leonardum de BasUea ei GuHditmm d$ P^^i^\ but witiiout Ae dedi-
cation to Mocenigo who had died in the meantime (1485)^ If Cam-
panus added anything of his own, his additions are at all events not
distinguished by any difference of t^pe or oAerwise; the enunciations
are in large type, and the rest is prmted continuously in smaller type.
There are no superscriptions to particular passages such as Ettduks
ex Campano^ Campanus^ Cm^ami addiiio^ or Omi$mU amMtoHo^ which
are found for the first time in the Paris edition of 1516 giving
both Campanus' version and that of Zamberti Qwesently to be men-
tioned).

1 501. G. Valla included in his enc3^opaedic work De expiUndis
et fugiendis rebus published in this year at Venice (iVk eudibus Atdi
Romant) a number of propositions with proofs and scholia translated
from a Greek MS. which was once in hiis possession (cod Mutin. ni
B, 4 of the isthc).

1 505. In this year Bartolomeo Zamberti (Zambertus) brought out
at Venice the first translation, from the Greek text, of the whole of tlie
Elements. From the titled as well sis from his prefaces to the CaUptriem
and DaUif with their allusions to previous translators ** who take some
things out of authors, omit some, and change some," or ** to that most
barl^rous translator " who filled a volume purporting to be Euclid's
''with extraordinary scarecrows, nightmares and phantasies," one object
of Zamberti's translation b dear. His animus against Campanus
appears also in a number of notes, e.g. when he condemns the terms
''helmuain" and '^ helmuariphe "* used by Campanus as barbarous^
un-Latin etc, and when he is roused to wrath by Campanus' unfortu- I
nate mistranslation of v. Def. 5. He does not seem to have had the >
penetration to see that Campanus was translating from an Arabic,
and not from a Greek, text Zamberti tells us that he spent
seven years over his translation of the thirteen Books of the
Elements. As he seems to have been bom in 1473, and the Elements
were printed as early as 1500, though the complete work (including the
Phaenomena^ Optica^ Catoptrica^ pata etc.) has the date 1505 at the
end, he must have translated Euclid before the age of 3a Heiberg
has not been able to identify the MS. of the Elements which Zamberti
used ; but it is clear that it belonged to the worse class of MSS., since
it contains most of the interpolations of the Theonine variety. Zam-
berti, as his title shows, attributed th^ proofs to Theon.

1509. As a counterblast to Zamberti, Luca Paciuolo brought out
an edition of Euclid, apparently at the expense of Ratdolt, at Venice f
{per Pagamnum de Paganinii), in which he set himself to vindicate
dampanus. The title-page of this now very rare edition' b^ns thus : |
*'The works of Euclid of Megara, a most acute philosopher and without '

^ The title begins thus: *'Eiiclidis megaresis philosophi platonicj mathematicaniml
discipliiuurum Janitoris : Habent in hoc volumine quicunque ad matbematicaro substantiam .
aspirant : elementomm libros xiij cum expositione Theonis insignis mathematici. quibus,
multa quae deerant ex lectione graeca sumpta addita sunt nee non plarima peniena et
praepostere: voluta in Campani interpretatione : ordinata digesta et castigata sant etc."
For a description of the book see Weissenbom, p. is sqq.
. ' See Weissenbom, p. 30 sqq.

CH. viii] TRANSLATIONS AND EDITIONS 99

question the chief of all mathematicians, translated by Campanus their
most faithful interpreter'' It proceeds to say that the translation had
been, through the fault of copyists, so spoiled and deformed that it
could scarcely be recognised as Euclid. Luca Paciuolo accordingly
has polished and emended it with the most critical judgment, has
corrected 129 figures wrongly drawn and added others, besides supply-
ing short explanations of difficult passages. It is added that Scipio
V^ius of Milan, distinguished for his knowledge '' of both languages'*
(le. of course Latin and Greek), as well as in medicine and the more
sublime studies, had helped to make the edition more perfect. Though
Zamberti is not once mentioned, this latter remark must have refer-
ence to Zamberti's statement that his translation was from the Greek
text ; and no doubt Zamberti is aimed at in the wish of Paciuolo's
" that others too would seek to acquire knowledge instead of merely
showing off, or that they would not try to make a market of the
things of which they are ignorant, as it were (selling) smoke*."
Weissenborn observes that, while there are many trivialities in Paci-
uolo's notes, they contain some useful and practical hints and explana-
tions of terms, besides some new proofs which of course are not
difficult if one takes the liberty, as Paciuolo does, of diverging from
Euclid's order and assuming for the proof of a proposition results not
arrived at till later. Two not inapt terms are used in this edition to
describe the figures of ill. 7, 8, tihe former of which is called the
gooses foot {pes anseris\ the second the peacock's tail {catula pavonis),
Paciuolo as the castigator of Campanus' translation, as he calls himself,
failed to correct the mistranslation of v. Def 5'. Before the fifth
Book he inserted a discourse which he gave at Venice on the
15th August, 1508, in S. Bartholomew's Church, before a select
audience of 500, as an introduction to his elucidation of that Book.
1 5 16. The first of the editions giving Campanus' and Zamberti's
translations in conjunction was brought out at Paris (i« offidna Henrici
Stephani e regione scholae Decretorum). The idea that only the enun-
ciations were Euclid's, and that Campanus was the author of the proofs
in his translation, while Theon was the author of the proofs in the Greek
text, reappears in the title of this edition; and the enunciations of the
added Books xiv., xv. are also attributed to Euclid, Hypsicles being
credited with the proofs'. The date is not on the title-page nor at the

^ "Atque utinam et alii cognoscere vellent nod ostentare ant ea quae nesdunt velati
fumum venditare non conarentar.'*

' Campanus' translation in Ratdolt's edition is as follows: "Quantitates quae dicnntur
oontinuam habere proportionaliutem, sunt, quarum equ^ multiplicia aut equa sunt aut
eou^ sibi sine interruptione addunt aut minuunt " ( !), to which Campanus adds the note :
^ Continue proportionalia sunt quorum omnia multiplicia equalia sunt continue proportionalia.
Sed noluit ipsam diffinitionem proponere sub hac forma, quia tunc difiiniret idem per idem,
aperte (? a parte) tamen rei est istud cum sua diffinitione convertibile."

' **£uclidis Megarensis Geometricorum Elementorum Libri xv. Campani Galli trans-
alpini in eosdem commentariorum libri xv. Theonis Alexandrini Bartholomaeo Zambertd
Veneto interprete, in tredecim priores, commentationum libri xiii. Hypsiclis Alexandrini in
duos posteriores, eodem Bartholomaeo Zamberto Veneto interprete, commentariorum libri n."
On the last page (161) is a similar sutement of content, but with the difference that the
expression **ex Campani... deinde Theonis... et Hypsiclis... /nu/iM^t^Kr." For description
see Weissenborn, p. 56 sqq.

7-^2

loo INTRODUCTION \ ' [€& vm

end, but the letter of dedication to Franjois Briconnet by Jacques
Leftvre is dated the day after the Epiphany, 151& The figures are
in the margin. The arrangement of the propositions is as follows :
first the enunciation with the beading Eudides ex Campano^ then the
proof with the note Campanus^ and after that, as Campani additio^ any
passage found in the ^ition of Campanus' translation but not in the
Gr^k text ; then follows the text of the enunciation translated from
the Greek with the heading Eudides ^x Zambtrto, and lastly the proof
headed Theo ex ZambertQ, There are separate figures for the two proofs.
This edition was reissued with few changes in 1537 and 1546 at Basel
(fipud lokaumm H€rvagium\ but ivith the addition of the Pkojenontitta^
Optica^ Catoptrka etc For the edition of 1537 the Paris edition of
1516 was collated with *'a Greek copy" (as the preface says) by
Christian Herlinj professor of mathematical studies at Strassburg,
who however seems to have done no more than correct one or two
passages by the help of the Basel editw princeps (1533), and add the
Greek word in cases where Zamberti's translation of it seemed unsuit-
able or inaccurate. '

We now come to # , t,

. . T
IL Editions of the Greek text, » •*

1 533 is the date of the editw prinaps, the title-page of which reads
as follows:

ETKAEIAOT STOIXEinN BIBA>- lEK

EK TON eEONOZ 1TN0Y2I0R
£«'? ToO avTov TO TrpmroVf e^^^dnop TlpoicXov 0ifi\* o*

Adiecta praefatiuncula in qua de disctplinis #

Mathematicis nonnihiL

BASILEAE APVD lOAN, HERVAGIVM ANNO

M^aXXXIlL MENSE SEPTEMBRL

The editor was Simon Grynaeus the elder (d* 1541), who, after
working at Vienna and Ofen, Heidelberg and Tubingen, taught last
of all at Basel, where theology was his main subject. His "prae-
fatiuncula" is addressed to an Englishman, Cuthbert Tonstall (1474-
1559)9 ^ho> having studied first at Oxford, then at Cambridge, where
he became Doctor of Laws, and afterwards at Padua, where in addi-
tion he learnt mathematics — mostly from the works of Regiomontanus
and Paciuolo — wrote a book on arithmetic* as **a farewell to the
sciences," and then, entering politics, became Bishop of London and
member of the Privy Council, and afterwards (i 530) Bishop of Durham.
Grynaeus tells us that he used two MSS. of the text of the Elements,
entrusted to friends of his, one at Venice by " Lazarus Bayfius "
(Lazare de BaTf, then the ambassador of the King of France at Venice),
the other at Paris by " loann. Rvellius " (Jean Ruel, a French doctor
and a Greek scholar), while the commentaries of Proclus were put at

^ De arti tufputamdiiiM qmatuor.

1

CH. viii] TRANSLATIONS AND EDITIONS ..;v loi

the disposal of Grynaeus himself by "loann. Claymundus** j;t*Qxford.
Heiberg has been able to identify the two MSS. used for* the** text ;
they are (i) cod. Venetus Marcianus 301 and (2) cod. Paris. g^.*2343
of the 1 6th c, containing Books I. — XV., with some scholia wHicJ^* are
embodied in the text. When Grynaeus notes in the margfff \t^e
readings from "the other copy," this "other copy" is as a rule** the
Paris MS., though sometimes the reading of the Paris MS. is taken,
into the text and the " other copy " of the margin is the Venice Mfi*
Besides these two MSS. Grynaeus consulted Zamberti, as is shown by^*
a number of marginal notes referring to " Zampertus " or to " latinum '
exemplar" in certain propositions of Books IX. — XL When it is con-
sidered that the two MSS. used by Grynaeus are among the worst, it
is obvious how entirely unauthoritative is the text of the editio prinups.
Yet it remained the source and foundation of later editions of the
Greek text for a long period, the editions which followed being
designed, not for the purpose of giving, from other MSS., a text more
nearly representing what Euclid himself wrote, but of supplying a
handy compendium to students at a moderate price.

1536. Orontius Finaeus (Oronce Fine) published at Paris {apud
Simonetn Colinaeuni) "demonstrations on the first six books of Euclid's
elements of geometry," " in which the Greek text of Euclid himself is
inserted in its proper places, with the Latin translation of Barth.
Zamberti of Venice," which seems to imply that only the enunciations
were given in Greek. The preface, from which Kastner quotes S says
that the University of Paris at tfiat time required, from all who
aspired to the laurels of philosophy, a most solemn oath that they
had attended lectures on the said first six Books. Other editions of
Fine's work followed in 1544 and 1551.

1545. The enunciations of the fifteen Books were published in
Greek, with an Italian translation by Angelo Caiani, at Rome {apud
Antonium Bladum Asulanum), The translator claims to have cor-
rected the books and " purged them of six hundred things which did
not seem to savour of the almost divine genius and the perspicuity of
Euclid'."

1 549. Joachim Camerarius published the enunciations of the first
six Books in Greek and Latin (Leipzig). The book had a preface by
Rhaeticus, a pupil of Copernicus, bom at Feldkirch in the Vorarlberg
1514, died 1576. Another edition with proofs of the propositions
of the first three Books was published by Moritz Steinmetz in 1577
(Leipzig).

1550. loan. Scheubel published at Basel (also per loan. Her-
vagiunt) the first six Books in Greek and Latin " together with true
and appropriate proofs of the propositions, without the use of letters "
(to denote points in the figures, the various straight lines and angles
being described in words*).

1557 (also 1558). Stephanus Gracilis published another edition
(repeated 1 573, 1 578, 1 598) of the enunciations (alone) of Books I. — xv.

* Kiistner, I. p. a6o. ' Heiberg, vol. v. p. cvii. * K&stner, i. p. 359.

%

...... INTRODUCTION i W^[ch. viii

in Grcetf^tid Latin at Paris (afiud GuUelmum Caviilat\ He remarks
tn thtf'preTace that for want of time he had changed scarcely anything
/in 5doJcs L — VI., but m the remaining Books he had emended what
[seemed obscure or inelegant in the Latin translation, while he had
' ad6pCed in its entirety the translation of Book X. by Pierre Mondor<i
{^P^lrus Montaureus), published separately at Paris in 1551- Gracilis
also added a few ** scholia."

**; 1564, In this year Conrad Dasypodius (Rauchfuss), the inventor
and maker of the clock in Strassburg cathedral, similar to the present
one, which did duty from 1571 to 1789, edited (Strassburg, Chn
Mylius) (i) Book L of the Elements in Greek and Latin with scholia,
(2) Book IL in Greek and Latin with Barlaain*s arithmetical version
of Book IL, and (3) the muncmticns of the remaining Books HI. — XllL
Book L was reissued with "vocabula quaedam geometrica" of Heron,
the enunciations of all the Books of the Eiemenis, and the other works
of Euclid, all in Greek and Latin. In the preface to (l) he says that it
had been for twenty-six years the rule of his school that all who were
promoted from the classes to public lectures should learn the first
Book, and that he brought it out, because there were then no longer
any copies to be had, and in order to prevent a good and fruitful
regulation of his school from falling through. In the preface to the
edition of 1571 he says that the first Book was generally taught in all
gymnasia and that it was prescribed in his school for the first class.
In the preface to (3) he tells us that he published the enunciations of
Books IIL — XIII, in order not to leave his work unfinished, but that, as
it would be irksome to carry about the whole work of Euclid in
extenso, he thought it would be more convenient to students of
geometry to learn the Eiefnmts if they were compressed into a smaller
book.

t62a Henry Briggs (of Briggs* logarithms) published the first
six Books in Greek with a Latin translation after Command in us,
"corrected in many places" (London, G. Jones).

1703 is the date of the Oxford edition by David Gregory which,
until the issue of Heiberg and Menge's edition, was still the only
edition of the complete works of Euclid ^ In the Latin translation
attached to the Greek text Gregory says that he followed Comman-
dinus in the main, but correct^ numberless passages in it by means
of the books in the Bodleian Library which belonged to Edward
Bernard (1638- 1696), formerly Savilian Professor of Astronomy, who
had conceived the plan of publishing the complete works of the ancient
mathematicians in fourteen volumes, of which the first was to contain
Euclid's Elements L — KV» As regards the Greek text, Gregory tells us
that he consulted, as far as was necessary, not a few MSS. of the better
sort, bequeathed by the great Savile to the University, as well as the
corrections made by Savile in his own hand in the margin of the Basel
edition. He had the help of John Hudson, Bodley s Librarian, who

^ ETEABtAOT tA ZCIZOMBKA. EucHdis q^e sapersunt omnut. Ex re<^nnotie
D«TidU Grcgorii M.D, Astronomiae Professoris SavilUnt ci R.S,S, Oxoniac, c Tbeairo
Sheldoniuto, An. Dom. iiDcciti.

I

CH. viii] TRANSLATIONS AND EDITIONS 103

punctuated the Basel text before it went to the printer, compared the
Latin version with the Greek throughout, especially in the Elements
and Data, and, where they differed or where he suspected the Greek text,
consulted the Greek MSS. and put their readings in the margin if
they agreed with the Latin and, if they did not agree, affixed an
asterisk in order that Gregory might judge which reading was geo-
metrically preferable. Hence it is clear that no Greek MS., but the
Basel edition, was the foundation of Gregory's text, and that Greek
MSS. were only referred to in the special passages to which Hudson
called attention.

1 8 14-18 1 8. A most important step towards a good Greek text
was taken by F. Peyrard, who published at Paris, between these years,
in three volumes, the Elements and Data in Greek, Latin and French ^
At the time (1808) when Napoleon was having valuable MSS. selected
from Italian libraries and sent to Paris, Peyrard managed to get two
ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat.
204 was also at Paris at the time, but all three were restored to their
owners in 18 14). Peyrard noticed the excellence of Cod. Vat. 190,
adopted many of its readings, and gave in an appendix a conspectus
of these readings and those of Gregory's edition ; he also noted here
and there readings from Vat. .1038 and various Paris MSS. He there-
fore pointed the way towards a better text, but committed the error
of correcting the Basel text instead of rejecting it altogether and
starting afresh.

1 824- 1 825. A most valuable edition of Books I. — ^vi. is that of
J. G. Camerer (and C. F. Hauber) in two volumes published at
Berlin". The Greek text is based on Peyrard, although the Basel
and Oxford editions were also used. There is a Latin translation
and a collection of notes far more complete than any other I have
seen and well nigh inexhaustible. There is no editor or commentator
of any mark who is not quoted from ; to show the variety of important
authorities drawn upon by Camerer, I need only mention the following
names : Proclus, Pappus, Tartaglia, Commandinus, Clavius, Peletier,
Barrow, Borelli, Wallis, Tacquet, Austin, Simson, Playfair. No words
of praise would be too warm for this veritable encyclopaedia of
information.

1825. J. G. C. Neide edited, from Peyrard, the text of Books
I. — VI., XI. and XII. {Halts Saxoniae).

1826-9. The last edition of the Greek text before Heiberg's is
that of E. F. August, who followed the Vatican MS. more closely
than Peyrard did, and consulted at all events the Viennese MS.
Gr. 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains
Books I. — XIII.

^ Etulidis quae iupersunt, Les (Euvres cTEuclide^ en Grec, en Laiin et en Fran^ais
d^aprls un manuserit tris-ancien, qui itait restiinconnu jusqu^h nos jours. Par F. Peyrard.
Ouvrage approuv^ par TlDStitut de France (Paris, chez M. Patris).

* Euclidis elementorum Ubri sex pricres graece et laiine commentarU e scriptis veterum ac
ncentiorum mathematuorum et PJUuieren maxime i/Zf/j/m/f (Berolini, sumptibus G. Reimeri).
Tom. I. 1834; torn. n. 1835.

I

«4 INTRODUCTION r [ciu vui

III, Latin versions or commentaries after 153?*"""

1545, Petrus Ramus (Pierre de la Ratnte* 15 15-1 572) is credited
with a translation of Euclid which appeared in 1545 and again in
1 549 at Parish Ramus, who was more rhetorician and logician thanj
geometer, also published in his ScAolae ma fA^malicae {i ^$g,Fr^nk{uTtl\
1569, Basel) what amounts to a series of lectures on Euclid's Ekmtnts^^
in which he criticises Euclid's arrangement of his propositions, the;
definittons, postulates and axioms^ all from the point of view of logiaj

1557. Demonstrations to the geometrical Elements of Euclid, six'
Books, by Peletarius (Jacques Peleticr), The second edition (1610)
contained the same with the addition of the "Greek text of Euclid**;
but only the enunciatmis of the propositions, as well as the defini-
tions etc*j are given in Greek (with a Latin translation), the rest is
in Latin only* He has some acute observations, for instance about
the "angle" of contact \

1S59» Johannes Buteo, or Borrel (1492-1572), published in anj
appendix to his book De quadraiura circuli some notes *'on the errors^
of Campanus, Zambertus, Orontius, Peletarius^ Pena, interpreters of
Euclid/* Buteo in these notes proved, by reasoned argument based
on original authorities, that Euclid himself and not Theon was the
author of the proofs of the propositions*

1566. Franciscus Flussates Candalla (Francois de Foix, Comte de
Candale, 1502-1594) "restored" the fifteen Books, following, as he
says, the terminology of Zamberti s translation from the Greek, but
drawing, for his proofs, on both Campanus and Theon (i,e, Zamberti)
except where mistakes in tliem made emendation necessary. Other
editions followed in 1578, 1602, 1695 (in Dutch).

1572. The most important Latin translation is that of Com-
mandinus (i 509-1 575) of Urbino, since it was the foundation of most
translations which followed it up to the time of Peyrard» including
that of Simson and therefore of those editions, numerous in England,
which give Euclid "chiefly after the text of Simson." Simson's first
(Latin) edition (1756) has ''ex versione Latina Federici Commandini"
on the title-page. Commandinus not only followed the original Greek
more closely than his predecessors but added to his translation some
ancient scholia as well as good notes of his own. The title of his
work is

Euclidis elementorum libri XV, una cum scholiis antiquis.
A Federico Comtnandino Urbinate nuper in latinum conversi,
commentariisque quibusdam iUustrati (Pisauri, apud CamiUum
Francischinum). I

He remarks in his preface that Orontius Finaeus had only edited '
six Books without reference to any Greek MS., that Peletarius had 1
followed Campanus' version from the Arabic rather than the Greek 1
text, and that Candalla had diverged too far from Euclid, having
rejected as inelegant the proofs given in the Greek text and 1
substituted faulty proofs of his own. Commandinus appears to have

1

Described by Boiicom|Migni, BulkUmo^ II. p. 389.

I

CH. viii] TRANSLATIONS AND EDITIONS 105

used, in addition to the Basel editio princeps^ some Greek MS., so far
not identified ; he also extracted his *' scholia antiqua " from a MS.
of the class of Vat. 192 containing the scholia distinguished by
Heiberg as "Schol. Vat" New editions of Commandinus' translation
followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and
Stone), 1756 (Books I. — vi., xi., xil. in Latin and English, by Simson),
1763 (Keill). Besides these there were many editions of parts of the
whole work, e.g. the first six Books.

1574. The first edition of the Latin version by Clavius*
(Christoph Schlussel, born at Bamberg 1537, died 161 2) appeared
in 1574, and new editions of it in 1589, 1591, 1603. 1607, 1612. It is
not a translation, as Clavius himself states in the preface, but it
contains a vast amount of notes collected from previous commentators
and editors, as well as some good criticisms and elucidations of his
own. Among other things, Clavius finally disposed of the error by
which Euclid had been identified with Euclid of Megara. He speaks
of the diflferences between Campanus who followed the Arabic
tradition and the " commentaries 01 Theon," by which he appears to
mean the Euclidean proofs as handed down by Theon ; he complains
of predecessors who have either only given the first six Books, or
have rejected the ancient proofs and substituted worse proofs of their
own, but makes an exception as regards Commandinus, ''a geometer
not of the common sort, who has lately restored Euclid, in a Latin
translation, to his original brilliancy." Clavius, as already stated, did
not give a translation of the Elements but rewrote the proofs, com-
pressing them or adding to them, where he thought that he could
make them clearer. Altogether his book is a most useful work.

1 62 1. Henry Savile's lectures {Praelectiones tresdecim in prin-
cipium Elenuntorum Euclidis Oxoniae habitae MDC.XX., Oxonii 162 1),
though they do not extend beyond I. 8, are valuable because they
grapple with the difficulties connected with the preliminary matter,
the definitions etc., and the tacit assumptions contained in the first
propositions.

1654. Andr6 Tacquet's Elementa geametriae planae et solidae
containing apparently the eight geometrical Books arranged for
general use in schools. It came out in a large number of editions up
to the end of the eighteenth century.

1655. Barrow's Euclidis Elementorum Libri XV breviter demon--
strati is a book of the same kind. In the preface (to the edition of
1659) he says that he would not have written it but for the fact that
Tacquet gave only eight Books of Euclid. He compressed the work
into a very small compass (less than 400 small pages, in the edition
of 1659, for the whole of the fifteen Books and the Data) by abbre-
viating the proofs and using a large quantity of symbols (which, he
says, are generally Oughtr^'s). There were several editions up to
1732 (those of 1660 and 1732 and one or two others are in English).

1 Euclidis eUmmtot-um lUni XV. Accessit xvi. de solidorum reguiarium comparatione,
Omtus terspicuis demonstraHambuSt accuratisque sektdiis Ulustrati, Aucicre Ckristophoro
Clamo (Romae, apud Vincentiam Accoltum), 1 vols.

io6 INTRODUCTION [ch. viii

1658. Giacomo Alfonso Borelli (1608-1679) published EmMks
resHtutus^ on apparently similar lines, which went Ummgh time mofe
editions (one in Italian, 1663).

1660. Claude Franfois Milliet Dechales' eight geometrical Books
of Euclid's Elements made easy. Dechales' versions of the Etemmts
had great vogue, appearing in French, Italian and English as wdl
as Latin. Riccardi enumerates over twenty editions.

1733. Sacdieri's EucUdes ab om$$i maeva vimUcatms swe emuOus
geometricus quo stabiliuntur prima ^sa gmmutrieie frimci^ is
important for his elaborate attempt to prove the parallel-postulate
forming an important stage in the history of the demopment of non-
Euclidean geometry.

i7S6- Simson's first edition, in Latin and in English. The Latin
title is

Euclidis elementorum libri prions Mr, Uim mmd eeim us ei du0-
decimus, ex versiotie latina Pederid Cammandini; sublaiis Us
quibus olim Ubri hi a Tkeone^ aUisve, vitiaH sumi, ei quiiusdam
Euclidis demonstrationibus resHiuiis. A Roberto SAmsom Jf.D.
Glasguae, in aedibus Academids excuddMUit Robertus et Andreas
Foulis, Academiae typografdiL

1802. Euclidis elementorum Ubri prims xn ex Com$nandimi ei
Gregorii versionibus latinis. In usmn iuventuOs Aeademicae...hy
Samuel Horsley, Bishop of Rochester. (Oxft>rd, Clarendcm Press.)

IV. Italian versioms or oomcentaries.

1543. Tartaglia's version, a second edition of which was pub-
lished in 1565^ and a third in 1585. It does not appear that he used
anv Greek text, for in the edition of is6s he mentions as available
only ''the first translation by Campano," "the second made by
Bartolomeo Zamberto Veneto who is still alive," "the editions of
Paris or Germany in which they have included both the aforesaid
translations," and "our own translation into the vulgar (tongue)."

1575. Commandinus' translation turned into Italian and revised
by him.

161 3. The first six Books "reduced to practice" by Pietro
Antonio Cataldi, re-issued in 1620, and followed by Books vii. — ix.
(1621) and Book x. (1625).

1633. Borelli's Latin translation turned into Italian by Domenico
Magni.

1680. Euclide restituto by Vitale Giordano.

169a Vincenzo Viviani's Elementi piani e solidi di Eudide
(Book v. in 1674).

1 The title-ptge of the edition of 1565 is as follows : EuciuU Magurrmu phiUsoptU^ s§U
imirmbitUrt dMe seimiie imaiMsmtOieit diligmUmsnte tyusifUU^, ei alia inttgritH ridotu^ per U
degmo prefessore di tal seientie NicoU Teartaka Briseianp, stimdo U due traJottiam, eon una
empla etpositwme dtlU istaso traddtore di m$iwc aggiwUa, talmenU chiara^ ctu ogm wudiccre
ingegm*^ stmna la MOtilia, otter sujfragio di oUtnT altra uimtia emt facUiU jm empau a
poiirU imUmdtn^ In Veneda, Appresio Cmtio Troiano, 1565.

^

CH. viii] TRANSLATIONS AND EDITIONS 107

173 1. EUnufiti geametrici piani e solidi di Euclide by Guido
Grandi. No translation, but an abbreviated version, of which new
editions followed one another up to 1806.

1749. Italian translation of Dechales with Ozanam*s corrections
and additions, re-issued 1785, 1797.

1752. Leonardo Ximenes (the first six Books). Fifth edition,
1819.

18 1 8. Vincenzo Flauti's Corso di geotnetria eUmentare e sublime
(4 vols.) contains (Vol. I.) the first six Books, with additions and a
dissertation on Postulate 5, and (Vol. II.) Books XL, XII. Flauti
also published the first six Books in 1827 and the Elements of geometry
of Euclid in 1843 and 1854.

V. German.

1558. The arithmetical Books vil.— ix. by Scheubel* (cf. the
edition of the first six Books, with enunciations in Greek and Latin,
mentioned above, under date 1550).

1562. The version of the first six Books by Wilhelm Holtzmann
(Xylander)*. This work has its interest as the first edition in German,
but otherwise it is not of importance. Xylander tells us that it was
written for practical people such as artists, goldsmiths, builders etc.,
and that, as the simple amateur is of course content to know facts,
without knowing how to prove them, he has often left out the proofs
altogether. He has indeed taken the greatest possible liberties with
Euclid, and has not grappled with any of the theoretical difficulties,
such as that of the theory of parallels.

165 1. Heinrich Hoffmann's Teutscher Euclides {2nd edition 1653),
not a translation.

1694. Ant Ernst Burkh. v. Pirckenstein's Teutsch Redender
Euclides (eight geometrical Books), "for generals, engineers etc."
"proved in a new and quite easy manner." Other editions 1699,

1744.

1697. Samuel Reyher's In teutscher Sprache vorgestellter Euclides
(six Books), "made easy, with symbols algebraical or derived from the
newest art of solution."

1 7 14. Euclidis XV Backer teutsch, "treated in a special and
brief manner, yet completely," by Chr. Schessler (another edition in
1729).

1773. The first six Books translated from the Greek for the
use of schools by J. F. Lorenz. The first attempt to reproduce
Euclid in German word for word.

1 78 1. Books XL, XII. by Lorenz (supplementary to the pre-
ceding). Also EuklicTs Elemente funfzehn Bucher translated from

^ Das sibendacht und nmnt buck des hochberiimbten Mathematici Euclidis Megarensis...
durch Magistrum Johann Sckeybl^ der loblichen univtrsiUt tu Tiibingaty des Euclidis und
Arithmclic Ordinaritn^ auss dem latein ins teutsch gebracht...,

• Die seeks erste Biicher Euclidis vom an fang oder grundder Geometry... Auss Griechiscker
sprach in die Teiitsck gebracht aigentlich erkldrt...Demcusen vormals in TeUtseher sprach nie
gueken warden... Durch Wilhelm HoUtman genani Xylander von Augspterg. Getruckht <a

io8 INTRODUCTION [cH. vui

the Greek by Lorenz (second edition 1798 ; editions of 1809^ 1818,
1824 by MoUweide, of 1840 by Dippe). The edition of 1834, and
I presume those before it, are shortened by the use of symbols and
the compression of the enunciation and ^setting-out" into one.

1807. Books I.— VL, Xl^ XIL ''newly transuited from the Greekp"
by J. K. F. HaufT.

1828. The same Books by Joh. Jos. Ign. Hoffmann ''as guide
to instruction in elementary geometry, followed in 1832 by obmrva-
tions on the text by the same editor.

1833. Die GeometrU des BukUd und das Wksm dirsMm by
E. S. Unger; also 1838, 185 1.

1901. Max Simon, Ettclid und dU Sichs planimetrischm Bikfur.

VI. French.

1564-1566. Nine Books translated by Pierre Forcadd, a pupil
and friend of P. de la Ramde.

1604. The first nine Books translated and annotated by Jean
Errard de Bar-le-Duc; second edition, 1605.

161 5. Denis Henrion's translation of the 15 Books (seven
editions up to 1676).

1639. The first six Books ^demonstrated by symtxds, by a
method very brief and intelligible^'' by Pierre Hdngone, mentioned
by Barrow as the only editor who, before him, had lued symbols for
the exposition of Euclid.

1672. Eight Books "rendus plus fiidles*' by Claude Francois
Milliet Dechales, who also brought out Les iUtnens d'Euclide ex-
pliquis d*un€ maniire nouvelle et trh facile^ which appeared in many .
editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), I
and was translated into Italian (1749 etc.) and English (by William ^
Halifax, 1685). 1

1804. In this year, and therefore before his edition of the Greek
text, F. Peyrard published the Elements literally translated into
French. A second edition appeared in 1809 ^^^ ^^ addition of the
fifth Book. As this second edition contains Books I. — vi. XI., XII.
and X. I, it would appear that the first edition contained Books I. — ^iv.,
VI., XL, XII. Peyrard used for this translation the Oxford Greek text
and Simson.

VII. Dutch.

1606. Jan Pieterszoon Dou (six Books). There were many later
editions. Kastner, in mentioning one of 1702, says that Dou explains
in his preface that he used Xylander's translation, but, having after-
wards obtained the French translation of the six Books by Errard
de Bar-le-Duc (see above), the proofs in which sometimes pleased
him more than those of the German edition, he made his Dutch
version by the help of both.

16 1 7. Frans van Schooten, "The Propositions of the Books of
Euclid's Elements'*; the fifteen Books in this version ''enlarged" by
Jakob van Leest in 1662.

1695. C. J. Voc^ht, fifteen Books complete, with Candalla's •' 16th.''

CH. viiij TRANSLATIONS AND EDITIONS 109

1702. Hendrik Coets, six Books (also in Latin, 1692); several
editions up to 1752. Apparently not a translation, but an edition for
school use.

1763. Pybo Steenstra, Books I. — Vl., XL, XII., likewise an abbre-
viated version, several times reissued until 1825.

1 VIII. English.

f 1570 saw the first and the most important translation, that of Sir

Henry Billingsley. The title-page is as follows :

THE ELEMENTS

OF GEOMETRIE

of the most auncient Philosopher

EVCLIDE

of Megara

Faithfully {now first) translated into the Englishe toung^
by H. Billingsley, Citizen of London. Whereunto are annexed
certaine Scholies^ Annotations^ and Inuentions^ of the best
Mathematidens, both of time pasty and in this our age.

With a very fruitfull Preface by M. I. Dee, specifying the
chiefe Mathematicall Scieces, what they are^ and whereunto
commodious: where^ also, are disclosed certaine new Secrets
Mathematicall and Mechanically vntill these our daies, greatly
missed.

Imprinted at London hy fohn Daye.

The Preface by the translator, after a sentence observing that with-
out the diligent study of Euclides Elementes it is impossible to attain
unto the perfect knowledge of Geometry, proceeds thus. " Wherefore
considering the want and lacke of such good authors hitherto in our
Englishe tounge, lamenting also the negligence, and lacke of zeale to
their countrey in those of our nation, to whom God hath geuen both
knowledge and also abilitie to translate into our tounge, and to

Ipublishe abroad such good authors and bookes (the chiefe instrumentes
of all leaminges): seing moreouer that many good wittes both of
gentlemen and of others of all degrees, much desirous and studious of
F these artes, and seeking for them as much as they can, sparing no
paines, and yet frustrate of their intent, by no meanes attaining to
that which they seeke : I haue for their sakes, with some charge and
great trauaile, faithfully translated into our vulgare touge, and set
abroad in Print, this booke of Euclide. Whereunto I haue added
easie and plaine declarations and examples by figures, of the defini-
tions. In which booke also ye shall in due place finde manifolde
I additions, Scholies, Annotations, and Inuentions: which I haue
gathered out of many of the most famous and chiefe Mathematicies,
both of old time, and in our age : as by diligent reading it in course,
ye shall well perceaue...."

It is truly a monumental work, consisting of 464 leaves, and there-
fore 928 pages, of folio size, excluding the lengthy preface by Dee.
The notes certainly include all the most important that had ever been

no INTRODUCTION [ck. vui

written, from those of the Greek commeiitators, Proclus and the others
whom he quotes, down to those of Dee himself on the last books.
Besides the fifteen Books, Billingsley included the ''sixteenth" added
by Candalla. The print and appearance of the book are worthy of its
contents ; and, in order that it may be understood how no pains were
spared to represent everything in the dearest and most perfect form, .
I need only mention that the figures of the propositions in Book XL J
are nearly all duplicated, one being the figure of Euclid, the other an 1
arrangement of pieces of paper (triangular, rectang^ular etc) pasted at
the edges on to the page of the book so that the pieces can be turned
up and made to show the real form of the solid figures represented.

Billingsley was admitted Lady Margaret Scholar of St John's
College, Cambridge, in 1 55 1, and he is also said to have studied at i
Oxford, but he did not take a d^ree at either University. He was ^
afterwards apprenticed to a London haberdasher and rapidly became
a wealthy merchant Sheriff of London in 1584, he was elected Lord
Mayor on 3 ist December, 1 596^ on the death, during his year of oflSce^
of Sir Thomas Skinner. From 1589 he was one of the Queen's four
" customers,'* or farmers of customs, of the port of London. In 1591
he founded three scholarships at St John's Collie for poor students,
and gave to the College for thdr noAintenance two messuages and
tenements in Tower Street and in Mark Lane, AllhallowSi Barking.
He died in 1606.

165 1. Elements of Geometry, The first VI Boocks: In a eompeti^
diousform contracted and demomtraUd by Captain Thomas Rudd, with
the mathematicall preface of John Dee (London).

1660. The first English edition of Barrow's Euclid (published in
Latin in 1655), appeared in London. It contained "the whole fifteen
books compendiously demonstrated"; several editions followed, in
1705, 1722, 1732, 1751.

1661. Eudids Elements of Geometry^ with a supplement of divers
Propositions and Corollaries, To which is added a Treatise of regular
Solids by Campane and Flussat ; likewise EuclicTs Data and Marinus
his Preface. Also a Treatise of the Divisions of Superficies^ ascribed to
Machomet Bagdedine^ but published by Commandine at the request of
f. Dee of London. Published by care and industry of John Leeke and
Geo. Serle, students in the Math. (London). According to Potts this
was a second edition of Billingsley's translation.

1685. William Halifax's version of Dechales' " Elements of Euclid i
explained in a new but most easy method " (London and Oxford).

1705. The English Euclide; being the first six Elements of I
Geometry^ translated out of the Greeks with annotations and useful/ (
supplements by Edmund Scarburgh (Oxford). A noteworthy and 4
useful edition. \

1708. Books L — ^VL, XL, XII., translated from Commandinus' Latin
version by Dr John Keill, Savilian Professor of Astronomy at Oxford.

Keill complains in his preface of the omissions by such editors as
Tacquet and Dechales of many necessary propositions (e.g. VL 27 — 29),
and of their substitution of proofs of their own for Euclid's. He praises
Barrow's version on the whole, though objecting to the ** algebraical "

CH. viii] TRANSLATIONS AND EDITIONS iix

form of proof adopted in Book ll., and to the excessive use of notes
and symbols, which (he considers) make the proofs too short and
thereby obscure: his edition was therefore intended to hit a proper
mean between Barrow's excessive brevity and Clavius' prolixity.

Keiirs translation was revised by Samuel Cunn and several times
reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782
the twelfth.

1 7 14. W. Whiston's English version (abridged) of The Elements
of Euclid with select theorems out of Archimedes by the learned Andr.
Tacguet.

1756. Simson's first English edition appeared in the same year as
his Latin version under the title :

The Elements of Euclid, viz, the first six Books together with
the eleventh and twelfth. In this Edition the Errors by which
Theon or others have long ago vitiated these Books are corrected mtd
some of Euclid's Demonstrations are restored. By Robert Simson
(Glasgow).

As above stated, the Latin edition, by its title, purports to be "ex
versione latina Federici Commandini,*' but to the Latin edition, as well
as to the English editions, are appended

Notes Critical and Geometrical; containing an Account of those
things in which this Edition differs from the Greek text; and ike
Reasons of the Alterations which have been made. As also Obser-
vations on some of the Propositions.

Simson says in the Preface to some editions (e.g. the tenth, of
1799) ^^^^ '*^^ translation is much amended by the friendly assistance
of a learned gentleman."

Simson's version and his notes are so well known as not to need
any further description. The book went through some thirty suc-
cessive editions. The first five appear to have been dated I7S6, 1762,
1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806,
the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844.
The DcUa '' in like manner corrected " was added for the first time in
the edition of 1762 (the first octavo edition).

1 78 1, 1788. In these years respectively appeared the two volumes
containing the complete translation of the whole thirteen Books by
James Williamson, the last English translation which reproduced
Euclid word for word. The title is

The Elements of Euclid, with Dissertations intended to assist
and encourage a critical examination of these Elements, as the most
effectual means of establishing a juster taste upon mathematical
} subjects than that which at present prevails. By James Williamson.

tin tfie first volume (Oxford, 1781) he is described as "MA.
Fellow of Hertford College," and in the second (London, printed by
T. Spilsbury, 1788) as "B.D." simply. Books v., vi. with the Con-
) elusion in the first volume are paged separately from the rest

1 78 1. An examination of the first six Books of Euclid's Elements,
by William Austin (London).

1795- John Playfair's first edition, containing "the first six Books
of Euclid with two Books on the Geometry of Solids." The book

INTRODUCTION [CB. Tin

a fifth edition in 1819, an eighth in i83l,a ninth in 1836^ and
Ith in 1846.

[826. Riccardi notes under this* date Euclid's EUmmis 9f Gta-

* containing the whole twelve Books translmtidinio Et^gKsktfr^m ike

ofPeyrard^ by George Phillips. The editor, who waft rasident

FQueens' College, Cambridge, 1857-1892, was bom in 1804 and

jltriculated at Queens' in 1826, so that he must have published the

3k as an undergraduate.

1828. A very valuable edition of the fiist six Books is diat of
l^ionysius Lardner, with commentary and ge om etrical exerdses, to
Irhich he added, in place of Books XL, XIL, a Treatise on Solid
reometry mostly based on L^endre. Lardner compresses the pro-
sitions by combining the enunciation and tihe setting-out, auid lie
[gives a vast number of riders and additional propositions in smaller
print The book had reached a ninth edition by 184^ and an eleventh
by 185s. Among other things, Lardner gives an Appendix ^on the
theory of parallel lines," in which he gives a short histoiy of the
attempts to get over the difficulty of the parallel-postulate^ down to
that of Legendre.

1833. T. Perronet Thompson's Gmmutvy wiUumt axioms^ or ike
first Book of EuclicTs Elements wiik aUerwOons emd fMes; emd em
intercalary book in which the straight Urn emd fleme are dnkfod from
properties of tlu sphere ^ with an appendix coniamit%g noOces of m e ikods
proposed for getting over the diffictUty in ike iwe^k euHam ofEneUd

Thompson (1783-1869) was 7th wrangler 1802, mklshipman 1803,
Fellow of Queens' College, Cambridge, 1804, and afterwards general
and politician. The book went through several editions, but» naving
been well translated into French by Van Tenac» is nid to have
received more recognition in France than at home.

1845. Robert Potts' first edition <(and one of the best) entitled:

Euclid's Elements of Geometry chiefly from the text of
Dr Simson with explanatory notes.. Jo which is prefixed an
introduction containing a brief outline of the History of Geometry.
Designed for tlu use of the higher forms in Public Schools a»id
students in the Universities (Cambridge University Press, and
London, John W. Parker), to which was added (1847) ^^
Appendix to the larger edition of Euclid s Elements of Geometry^
containing additional notes on the Elements, a short tract on trans-
versalSy and hints for the solution of the problems etc.
1862. Todhunter's edition.

The later English editions I will not attempt to enumerate ; their
name is l^ion and their object mostiy that of adapting Euclid for school
use, with all possible gradations of departure from his text and order.

IX. Spanish.

1576. The first six Books translated into Spanish by Rodrigo
^amorano.

1637. The first six Books translated, with notes, by L. Carduchi.

1689. Books L — VL, XL, xiL, translated and explained by Jacob
Knesa.

1

CH. viii] TRANSLATIONS AND EDITIONS 113

X. Russian.

1739. Ivan Astaroff (translation from Latin).

1789. Pr. SuvorofTand Yos. Nikitin (translation from Greek).

1 880. Vachtchenko-Zakhartchenko.

(18 1 7. A translation into Polish by Jo. Czecha.)

XI. Swedish.

1744. Mirten Stromer, the first six Books; second edition 1748.
The third edition (1753) contained Books XI. — xii. as well; new
editions continued to appear till 1884.

1836. H. Falk, the first six Books.

1844, J 84s, 1859. P. R. BrSkenhjelm, Books I. — VI., XI., XII.

i8sa F. A. A. Lundgren.

1850. H. A. Witt and M. E. Areskong, Books I.— vi., xi., xil.

XII. Danish.

174s. Ernest Gottlieb Ziegenbalg.
1803. H. C. Linderup, Books I. — VI.

XIII. Modern Greek.
1820. Benjamin of Lesbos.

H. E.

-*■ -i^-

-t.V* ,Hr >

CHAPTER IX.

{ 1, ON THE NATURE OF ELEMENTS.

It would not: be easy to find a more lucid explanation of the terms
tknunt and eUmentafy, and of the distinction between them, than
is found in Proc1us\ who is doubtless, here as so often, quoting
from Geminus, There are, says Proclus, in the whole of geometry
certain leading theorems, bearing to those which follow the relation of I
a principle, all- pervading, and furnishing proofs of many properties*
Such theorems are called by the name of tlemintsx and their function
may be compared to that of the letters of the alphabet in relation to
language, letters being indeed called by the same name in Greek

The term elententaty, on the other hand, has a wider application :
it is applicable to things "which extend to greater multiplicity, and,
though possessing simplicity and elegance, have no longer the same
dignity as the eUments^ because their investigation is not of general
use in the whole of the science, eg, the proposition that in triangles
the perpendiculars from the angles to the trans vei^e sides meet in a
point."

" Again, the term element is used in two senses, as Menaechmus
says. For that which is the means of obtaining is an element of that
which is obtained, as the first proposition in Euclid is of the second,
and the fourth of the fifth. In this sense many things may even be
said to be elements of each other, for they are obtained from one
anotiier. Thus from the fact that the exterior angles of rectilineal
figures are (together) equal to four right angles we deduce the number
of right angles equal to the internal angles (taken together)*, and
via versa. Such an element is like a lemma. But the term element is
otherwise used of that into which, being more simple, the composite is
divided ; and in this sense we can no longer say that everything is an
element of everything, but only that things which are more of the ^
nature of principles are elements of those which stand to them In the
relation of results, as postulates are elements of theorems. It is

^ Prodos, Comm, on Etui, i., ed. Friedleio, f>p. 7a sqq.

* T^ vX^of rfir hrrht 6p0tut f^wr. If the text is nght, we most apparently take it as ''the
number of the angles equal to right angles that there are inside,'* i.e. that are made up by
the internal angles.

I

CH. IX.J1] ON THE NATURE OF ELEMENTS 115

according to this signification of the term element that the elements
found in Euclid were compiled, being partly those of plane geometry,
and partly those of stereometry. In like manner many writers have
drawn up elementary treatises in arithmetic and astronomy.

*' Now it is difficult, in each science, both to select and arrange in
due order the elements from which all the rest proceeds, and into
which all the rest is resolved. And of those who have made the
attempt some were able to put together more and some less ; some
used shorter proofs, some extended their investigation to an indefinite
length ; some avoided the method of reductio ad absurdum^ some
2l\o\A^ proportion \ some contrived preliminary steps directed against
those who reject the principles ; and, in a word, many different
methods have been invented by various writers of elements.

" It is essential that such a treatise should be rid of everything
superfluous (for this is an obstacle to the acquisition of knowledge) ;
it should select everything that embraces the subject and brings it to
a point (for this is of supreme service to science) ; it must have great
r^ard at once to clearness and conciseness (for their opposites trouble
our understanding); it must aim at the embracing of theorems in
general terms (for the piecemeal division of instruction into the more
partial makes knowledge difficult to grasp). In all these ways
Euclid's system of elements will be found to be superior to the rest ;
for its utility avails towards the investigation of the primordial
figures*, its clearness and organic perfection are secured by the
precession from the more simple to the more complex and by the
foundation of the investigation upon common notions, while generality
of demonstration is secured by die progression through the theorem's
which are primary and of the nature of principles to the things sought.
! As for the things which seem to be wanting, they are partly to be
1 discovered by the same methods, like the construction of the scalene
• and isosceles (triangle), partly alien to the character of a selection of
elements as introducing hopeless and boundless complexity, like the
! subject of unordered irrationals which Apolionius worked out at
length', and partly developed from things handed down (in the
elements) as causes, like the many species of angles and of lines.
These things then have been omitted in Euclid, though they have
received full discussion in other works ; but the knowledge of them is
derived from the simple (elements)."

Proclus, speaking apparently on his own behalf, in another place
distinguishes two objects aimed at in Euclid's Elements. ZTEe first
has reference to the matter of the investigation, and here, li^e a good
Platonist, he takes the whole subject of geometry to be concerned
with the " cosmic figures," the five regular solids, which in Book XIIL

* r(a9 d^x^'f^ ^iiM^rciir, by which Proclus probably means the regular polyhedxa
(Tannery, p. i43»-)-

' We have no more than the most obscure indications of the character of this work in an

. Arabic MS. analysed by Woepcke, Essed d^une restitution de travaux perdm dAtolUmius

tur Us fuantith irrationelUs d*apris da indicaticHS tirits d*un manuscrit arabi in Mhnoires

fristntis h ttuadhnie da sciences^ XI v. 658—720, Paris, 1856. Cf. Cantor, Gtsch. d, Maik.

't* PP* 34^~~9 * details are also given in my notes to Book x.

i' 8-a

li

1 16 INTRODUCTION [CH. dl f i

are constructed, inscribed in a sphere and compared with one another.
The second object is relative to the learner; and, from this standpoint,

^the elements may be described as ''a means of perfecting the learnei's
understanding with reference to the whole of geometiy. | For, starting
from these (elements), we shall be able to acquire knowledge of the
other parts of this science as well, while without them it is impossible
for us to get a grasp of so complex a subject, and knowledge ci the
rest is unattainable. As it is, the theorems which are most of the
nature of principles, most simple, and most akin to the first hypotheses
are here collected, in their appropriate order; and the proon of all
other propositions use these theorems as thoroughly well known, and
start from them. Thus Archimedes in the books on the sphere and
cylinder, Apollonius, and all other geometers, clearly use the theorems

proved in this very treatise as constituting admitted principles\''

Aristotle too speaks of elements of geometry in the same sense.
Thus: "in geometry it is well to be thoroughly versed in the
elements*"; "in general the first of the elements are, given the
definitions, e.g. of a straight line and of a circle^ most easy to prove,
although of course there are not many data that can be used to
establish each of them because there are not many middle terms*";
"among geometrical propositions we call those 'elements' the proofs of
which are contained in the proofs of all or most of such propositions^*';
"(as in the case of bodies)^ so in like manner we speak of the elements

' of geometrical propositions and, generally, of demonstrations ; for the
demonstrations which come first and are attained in a variety of
other demonstrations are called elements of those demonstrations...
the term element is applied by analogy to that which, being one and
small, is useful for many purposes'."

§ 2. ELEMENTS ANTERIOR TO EUCLID'S.

The early part of the famous summary of Proclus was no doubt
drawn, at least indirectly, from the history of geometry by Eudemus ;
this is generally inferred from the remark, made just after the mention
of Philippus of Mende, a disciple of Plato, that "those who have
written histories bring the development of this science up to this
point" We have therefore the best authority for the list of writers of
elements given in the summary. Hippocrates of Chios (fl. in second
half of 5th c.) is the first ; then Leon, who also discovered diarismiy
put together a more careful collection, the propositions proved in it
being more numerous as well as more serviceable*. Leon was a little
older than Eudoxus (about 390-337 B.C) and a little younger than
Plato (429-348 B.C.), but did not belong to the latter's school. The

* Prodtts, pp. 70, 10 — 71, ai.

* T^s VIII. 14, 163 b as. » To^s viii. 3, 158 b 35. * Mett^k. 998 a as.

* ifaath, loi^a 35— b 5.

* Prodos, p. 66, ao iSrrt rdr Atforra iral tA 9roix«<a «wtft4Mu r^ rt rX^^i k9X rj %pd^
rQf9 dmnnf/thttif iwifUkkm^.

I

CH. ix.§a] ELEMENTS ANTERIOR TO EUCLID'S 117

geometrical text-book of the Academy was written by Theudius of
Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of
Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus
consorted together in the Academy and carried on their investigations
in common. Theudius " put together the elements admirably, making
many partial (or limited) propositions more generals" Eudemus
mentions no text-book after that of Theudius, only adding that Her-
motimus of Colophon "discovered many of the elements"." Theudius
then must be taken to be the immediate precursor of Euclid, and no
doubt Euclid made full use of Theudius as well as of the discoveries of
Hermotimus and all other available material. Naturally it is not in
Euclid's Elements that we can find much light upon the state of the
subject when he took it up ; but we have another source of informa-
tion in Aristotle. Fortunately for the historian of mathematics,
Aristotle was fond of mathematical illustrations ; he refers to a con-
siderable number of geometrical propositions, definitions etc., in a
way which shows that his pupils must have had at hand some text-
book where they could find the things he mentions; and this text-book
must have been that of Theudius. Heiberg has made a most valuable
collection of mathematical extracts from Aristotle*, from which much
is to be gathered as to the changes which Euclid made in the methods
of his predecessors ; and these passages, as well as others not included
in Heiberg's selection, will often be referred to in the sequel.

§3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES,
AND AXIOMS.

On no part of the subject does Aristotle give more valuable
information than on that of the first principles as, doubtless, generally
accepted at the time when he wrote. One long passage in the
Posterior Analytics is particularly full and lucid, and is worth quoting
in extenso. After laying it down that every demonstrative science
starts from necessary principles^ he proceeds':

^ By first principles in each genus I mean those the truth of which
It is not possible to prove. What is denoted by the first (terms) and
those derived from them is assumed ; but, as regards their existence,
this must be assumed for the principles but proved for the rest. Thus
what a unit is, what the straight (line) is, or what a triangle is (must
be assumed) ; and the existence of the unit and of magnitude must
also be assumed, but the rest must be proved. Now of Uie premisses
used in demonstrative sciences some are peculiar to each science and
others common (to all), the latter being common by analogy, for of
course they are actually useful in so far as they are applied to the sub-
ject-matter included under the particular science. Instances of first

^ Proclus, P' 67, 14 icoi 7dp rd rroixeSa Ka\(at avpira^tp koI roXKii rOif fitpucwp [6piKQp (?)
Friedlein] KaBoXucirtpa iwolyfatw.

' Proclus, p. 67, 33 Twv CTOixtif^ roXXd dptvp€.

* Maikemaiiscka mu AristoUies in Abhandlungen wur Gtsch, d, math. Wissenichafttn^
xvni. Heft (1904), pp. I— 49*

^ Anal, post, I. 6, 74 b 5. * ibitL i. 10, 76 a 31 — 77 a 4.

1 18 INTRODUCTION [ch. ul f 3

principles peculiar to a science are fhe assumptions that a Une is of
such and such a character, and similarly for the straight (line); whereas
it is a common principle, for instance, that, if equals be subtracted
from equals, the remainders are equal. But it is enough that each tA
the common principles is true so far as regards the particular genus
(subject-matter) ; for (in geometiy) the effect will be the same even if
the common principle be assumed to be true, not of everything, but
only of magnitudes, and, in arithmetic, of numbers.

'* Now the things peculiar to the science, the existence of which
must be assumed, are the things with r efer en ce to which the science
investigates the essential attributes, e.g. arithmetic with reference to
units, and geometry with reference to points and lines. With these
things it is assumed that they exist and that they are of such and
such a nature. But, with regard to their essential properties, what is
assumed is only the meaning of each term employed : thus arithmetic
assumes the answer to die question what is (meant by) 'odd* or
'even,' 'a square' or 'a cube,' and geometry to the question
what is (meant by) 'the irrational ' or 'deflection' or (the so-called)
'verging' (to a point); but that there are such things is proved by
means of the common principles and of what has already been
demonstrated. Similarly with astronomy. For every demonstrative
science has to do with three things, (1) tne things wUch are assumed
to exist, namely the genus (subject-matter) in ei^ case, the essential
properties of which the science investigates, (2) the common axioms
so-called, which are the prinuuy source of demonstration, and (3) the
properties with regard to which all that is assumed is the meaning of
the respective terms used. There is, however, no reason why some
sciences should not omit to speak of one or other of these things.
Thus there need not be any supposition as to the existence of the
genus, if it is manifest that it exists (for it is not equally clear that
number exists and that cold and hot exist) ; and, with regard to the
properties, there need be no assumption as to the meaning of terms if
it is clear : just as in the common (axioms) there is no assumption as
to what is the meaning of subtracting equads from equals, because it is
well known. But none the less is it true that there are three things
naturally distinct, the subject-matter of the proof, the things provod,
and the (axioms) from which (the proof starts).

•'Now that which is per se necessarily true, and must necessarily be
thought so, is not a hypothesis nor yet a postulate. For demon-
stration has not to do with reasoning from outside but with the
reason dwelling in the soul, just as is the case with the syllogism.
It is always possible to raise objection to reasoning from outside,
but to contradict the reason within us is not always possible. New
anything that the teacher assumes, though it is matter of proft,
without proving it himself, is a hypothesis if the thing assumed is
believed by the learner, and it is moreover a hypothesis, not abso-
lutely, but relatively to the particular pupil ; but, if the same thing
b assumed when the learner either has no opinion on the subject
or is of a contrary opinion, it is a postulate. This is the difference

CH. DL§3] FIRST PRINCIPLES 119

between a hypothesis and a postulate ; for a postulate is that which

is rather contrary than otherwise to the opinion of the learner, or

I whatever is assumed and used without being proved, although matter

Ffor demonstration. Now definitions are not hypotheses, for they do
not assert the existence or non-existence of anything, while hypotheses
are among propositions. Definitions only require to be understood :
a definition is therefore not a hypothesis, unless indeed it be asserted
f that any audible speech is a hypothesis. A hypothesis is that from
I the truth of which, if assumed, a conclusion can be established. Nor
i are the geometer's hypotheses false, as some have said : I mean those
^ who say that ' you should not make use of what is false, and yet the
I geometer falsely calls the line which he has drawn a foot long when
t it is not, or straight when it is not straight' The geometer bases no
' conclusion on the particular line which he has drawn being that which
he has described, but (he refers to) what is illustrated by the figures.
Further, the postulate and every hypothesis are either universal or
particular statements; definitions are neither" (because the subject
is of equal extent with what is predicated of it).

Every demonstrative science, says Aristotle, must start from in-
demonstrable principles : otherwise, the steps of demonstration would
be endless. Of these indemonstrable principles some are (a) common
to all sciences, others are {b) particular, or peculiar to the particular
science ; {a) the common principles are the axioms, most commonly
illustrated by the axiom that, if equals be subtracted from equals, the
remainders are equal. Coming now to (b) the principles peculiar to
the particular science which must be assumed, we have first the genus
or subject-matter, the existence of which must be assumed, viz. magni-
tude in the case of geometry, the unit in the case of arithmetic. Under
this we must assume definitions of manifestations or attributes of the
genus, e.g. straight lines, triangles, deflection etc. The definition in
itself says nothing as to the existence of the thing defined : it only
requires to be understood. But in geometry, in addition to the genus
and the definitions, we have to assume the existence of a kvf frimary
things which are defined, viz. points and lines only : the existence
of everything else, e.g. the various figures made up of these, as
triangles, squares, tangents, and their properties, e.g. incommensur-
ability etc., has to be proved (as it is proved by construction and
demonstration). In arithmetic we assume the existence of the unit:
but, as regards the rest, only the definitions, e.g. those of odd, even,
square, cube, are assumed, and existence has to he proved. We have then
clearly distinguished, among the indemonstrable principles, axioms
and definitions, A postulate is also distinguished from a hypothesis,
the latter being made with the assent of the learner, the former
iRthout such assent or even in opposition to his opinion (though,
strangely enough, immediately after saying this, Aristotle gives a
wider meaning to "postulate" which would cover "hypothesis" as well,
namely whatever is assumed, though it is matter for proof, and used
without being proved). Heiberg remarks that there is no trace in
Aristotle of Euclid's Postulates, and that " postulate" in Aristotle has

X 20 INTRODUCTION [ch. xx. f 3

a different meaning. He seems to base this on the alternative
description of postulate, indistinguishable from a hypothesis; but.
if we take the other description in which it is distinguished fiom a
hypothesis as being an assumption of something which is a proper
subject of demonstration without the assent or against the opinion of
the learner, it seems to fit Euclid's Postulates fainy well, not only the
first three (postulating three constructionsX but eminently also the other
two, that all right angles are equal, and that two straight lines meeting
a third and making the internal angles on the same side of it less than
two right angles will meet on that side. Aristotle's description also
seems to me to suit the ''postulates" mth which Archimedes b^ins
his book On the equilibrium ofplams, namely that equal weights baliuice
at equal distances, and that equal weights at unequal distances do not
balance but that the weight at the longer dbtance will prevail

Aristotle's distinction also between kfpothisis and definiiiaH^ and
between hypothesis and eua&m^ is clear from the foUowine passa^:
''Among immediate syllogistic principles, I call that a msis which
it is neither possible to prove nor essential for any one to hold who
is to learn anything ; but that which it is necessary for any one to
hold who is to learn anything whatever is an axiom : for there are
some principles of this kind, and that is the most usual name 1^
which we speak of them. But, of theses, one kind is that which
assumes one or other side of a predication, as, for instance^ that
something exists or does not exist, and this is a Ig^thisis ; the other,
which makes no such assumption, is a defimiioH. For a definition is
a thesis : thus the arithmetician posits (rlOrrai) that a unit is that
which is indivisible in respect of quantity ; but this is not a hypo-
thesis, since what is meant by a unit and the fact that a unit exists
are different things*."

Aristotle uses as an alternative term for axioms ''common (things),"
ra tcoivd, or "common opinions" (tcoipol So^ai), as in the following
passages. *' That, when equals are taken from equals, the remainders
are equal is (a) common (principle) in the case of all quantities, but
mathematics takes a separate department (diroXafiovaa) and directs its
investigation to some portion of its proper subject-matter, as e.g. lines
or angles, numbers, or any of the other quantities"." "The common
(principles), e.g. that one of two contradictories must be true, that
equals taken from equals etc, and the like*...." " Withr^[ard to the
principles of demonstration, it is questionable whether they belong to
one science or to several. By principles of demonstration I mean the
common opinions from which all demonstration proceeds, e.g. that one
of two contradictories must be true, and that it is impossible for the
same thing to be and not be^" Similarly "every demonstrative
(science) investigates, with r^ard to some subject-matter, the essential
attributes, starting from the common opinions*** We have then here,
as Heiberg says, a sufficient explanation of Euclid's term for axioms,

* Anai, post. I. a, 7a a 14—94. * Mttaph, 1061 b 19—34.

* Anai. past, 1. 11, 77 a 30. « Mtuipk. 996b 36—30.

* Mdapk. 997 a so— ^3.

CH. ix.§3] FIRST PRINCIPLES lai

viz. common notions {koivoX hpouti), and there is no reason to suppose
it to be a substitution for the original term due to the Stoics : cf.
Proclus* remark that, according to Aristotle and the geometers, axiom
and common notion are the same things

Aristotle discusses the indemonstrable character of the axioms
in the Metaphysics. Since "all the demonstrative sciences use the
axiomsV' the question arises, to what science does their discussion
belong'? The answer is that, like that of Being {ovala), it is the
province of the (first) philosopher^ It is impossible that there should
be demonstration of everything, as there would be an infinite series of
demonstrations : if the axioms were the subject of a demonstrative
science, there would have to be here too, as in other demonstrative
sciences, a subject-genus, its attributes and corresponding axioms* ; thus
there would be axioms behind axioms, and so on continually. The
axiom is the most firmly established of all principles*. It is ignorance
alone that could lead any one to try to prove the axioms' ; the supposed
proof would be a petitio principii*. If it is admitted that not every-
thing can be proved, no one can point to any principle more truly
indemonstrable*. If any one thought he could prove them, he could
at once be refuted ; if he did not attempt to say anything, it would
be ridiculous to argue with him : he would be no better than a
vegetable". The first condition of the possibility of any argument
whatever is that words should signify something both to the speaker
and to the hearer: without this there can be no reasoning with any one.
And, if any one admits that words can mean anything to both hearer
and speaker, he admits that something can be true without demon-
stration. And so on".

It was necessary to give some sketch of Aristotle's view of the
first principles, if only in connexion with Proclus' account, which is
as follows. As in the case of other sciences, so ''the compiler of
elements in geometry must give separately the principles of the
science, and after that the conclusions from those principles, not
giving any account of the principles but only of their consequences.
No science proves its own principles, or even discourses about them :
they are treated as self-evident... Thus the first essential was to dis-
tinguish the principles from their consequences. Euclid carries out
this plan practically in every book and, as a preliminary to the whole
enquiry, sets out the common principles of this science. Then he
divides the common principles themselves into hypotheses^ postulates^
and axioms. For all these are different from one another : an axiom,
a postulate and a hypothesis are not the same thing, as the inspired
Aristotle somewhere says. But, whenever that which is assumed and
ranked as a principle is both known to the learner and convincing in
itself, such a thing is an axiom^ e.g. the statement that things which
are equal to the same thing are also equal to one another. When, on

^ Proclus, p. 104, 8. * Mdaph. 907 a 10.

' ibid, 99(5 36. ^ ibid, 1005 a 31— b 11. * ibid. 997 a 5 — 8.

* ilnd. 1005 b II — 17. ' ibid. 1006 a 5. * ibid. 1006 a 17.

* fA«/. 1006a la ^ f!^f^ 1006a II— 15. " ftW. 1006 a 18 sqq.

&.

123 INTRODUCTION [cB. n. f 3

the other hand, the pupil has not the notion of what is told him
which carries conviction in itself, but nevertheless iBys it down and
assents to its being assumed, such an assumption is a Ig^thnis,
Thus we do not preconceive by virtue of a common notion, and
without being taught, that the circle is such and such a figure, but,
when we are told so, we assent without demonstration. When again
what is asserted is both unknown and assumed even without the
assent of the learner, then, he says, we call this a postidaU^ e.g. that
all right angles are equal This view of a postulate is clearly implied
by those who have made a spedal and systematic attempt to show,
with r^[ard to one of the postulates, that it cannot be assented to by
any one straight off. According then to the teaching of Aristotle, an
axiom, a postulate and a hypothesis are thus distinguished^**

We observe, first, that Proclus in this passage confuses kypotlksis
and definitions, although Aristotle had made the distinction quite
plain. The confusion may be due to his having in his mind a passage
of Plato from which he evidently got the phrase about ^ not giving
an account of" the principles. The passage is*: ^ I think you know
that those who treat of geometries and calculations (arithmetic) and
such things take for granted (Airotf^/Myoi) odd and even, figures,
angles of three kinds, and other things akin to these in each subject,
implying that they know these things, and, though using them as
hypotheses, do not even condescend to give any account of them
either to themselves or to others, but b^n from these things and
then go through everything else in order, arriving ultimatdy, l^
recognised methods, at the conclusion which they started in search
of." But the hypothesis is here the assumption, e.g. * that there may
be suck a thing as length without breadth, henceforward called a line*,'
and so on, without any attempt to show that there is such a thing ;
it is mentioned in connexion with the distinction between Plato's
'superior' and 'inferior' intellectual method, the former of which
uses successive hypotheses as stepping-stones by which it mounts
upwards to the idea of Good.

We pass now to Proclus' account of the difference between postu-
lates and axioms. He begins with the view of Geminus, according
to which " they differ from one another in the same way as theorems
are also distinguished from problems. For, as in theorems we propose
to see and determine what follows on the premisses, while in problems
we are told to find and do something, in like manner in the axioms
such things' are assumed as are manifest of themselves and easily
apprehended by our untaught notions, while in the postulates we
. assume such things as are easy to find and effect (our understanding
suffering no strain in their assumption), and we require no complication
of macluneryV'..."Both must have the characteristic of being simple

* Proclns, pp. 75, 10—77, *•

* RipubHc, VI. CIO c. CJf. Aristotle, Nic. Eth^ 1 151 a 17.

* H. }%c\aKmtjaurHal of Phiiohgy, vol. X. p. 144.

^ Produs, pp. 178, 13—179, 8. In Ulnstntion Prodos contrasts the drawins of a straight
line or a circle with the drawing of a *' single-torn spiral " or of an equilateral triangle, the

i

*"

CH. IX. §3] FIRST PRINCIPLES 123

and readily grasped, I mean both the postulate and the axiom ; but
the postulate bids us contrive and find some subject-matter (v\v) to
exhibit a property simple and easily grasped, while the axiom bids us
assert some essential attribute which is self-evident to the learner,
just as is the fact that fire is hot, or any of the most obvious things ^"

Again, says Proclus, '' some claim that all these things are alike
postulates, in the same way as some maintain that all things that are
sought are problems. For Archimedes begins his first book on /ft-
equilibrium^ with the remark ' I postulate that equal weights at equal
distances are in equilibrium,' though one would rather call this an
axiom. Others call them all axioms in the same way as some regard
as theorems everything that requires demonstration'."

" Others again will say that postulates are peculiar to geometrical
subject-matter, while axioms are common to all investigation which
is concerned with quantity and magnitude. Thus it is the geometer
who knows that all right angles are equal and how to produce in
a straight line any limited straight line, whereas it is a common notion
that things which are equal to the same thing are also equal to one
another, and it is employed by the arithmetician and any scientific
person who adapts the general statement to his own subjects"

The third view of the distinction between a postulate and an axiom
is that of Aristotle above described*.

The difficulties in the way of reconciling Euclid's classification
of postulates and axioms with any one of the three alternative views
are next dwelt upon. If we accept the first view according to which
an axiom has reference to something known, and a postulate to
something done, then the 4th postulate (that all right angles are
equal) is not a postulate ; neither is the 5th which states that, if a
straight line falling on two straight lines makes the interior angles
on the same side less than two right angles, the straight lines, if
produced indefinitely, will meet on that side on which are the angles
I less than two right angles. On the second view, the assumption that
two straight lines cannot enclose a space, "which even now," says
Proclus, "some add as an axiom," and which is peculiar to the
subject-matter of geometry, like the fact that all right angles are
equal, is not an axiom. According to the third (Aristotelian) view,
"everything which is confirmed {irurrovraC) by a sort of demonstration

spiral requiring more complex machinery and even the equilateral triangle needing a certain
method. ** For the geometrical intelligence will say that by conceiving a straight line fixed
at one end but, as regards the other end, moving round the fixed end, and^a point moving
along the straight line from the fixed end, I have described the single-turn spiral ; for the
end of the straight line describing a circle, and the point moving on the straignt Une simul-
taneously, when they arrive and meet at the same point, complete such a spiral. And again,
if I draw equal circles, join their common point to the centres of the circles and draw a

I straight line from one of the centres to the other, I shall have the equilateral triangle.

. These thinsfs then are far from being completed by means of a single act or of a moment's

' thought" (p. 180, 8— ai).

, ' Proclus, p. 181, 4— II.

* It is necessary to coin a word to render dycovpportwr, which is moreover in the plural.
The title of the treatise as we have it is Equilibria of planes or centres tf gravity of planes in
Book I and Equilibria of plana in Book ii.

» Proclus, p. 181, 16---13. * ibid. p. 183, 6—14. • Pp. 118, 119.

134 INTRODUCnOli fiaiL»t3

will be a postulate, and whaA h incapable of proof wUi be an aacicMii^*
This last statement of Produs is h>o8e, as regards the axdom, bacioie
it omits Aristotle's requirement that the axiom dioold be a iitf-
evident truth, and one that must be admitted by any one who la to
learn anything at all, and, as rq^ards the postulate, because Afistade
calls a postulate something assumed without proof Ihou^i it is
"matter of demonstration" ((kweUtierim Jh% but says nothing ofim
^t^ji-demonstration of the postulates. CHi the whme I think it is
from Aristotle that we get the best idea of what Eudid undtnUmd
by a postulate and an axiom or common nolioii. Thus ArisMttfs
account of an axiom as ai prindpie common to all scienoesi wUdi b
self-evident, though incapaUe of proo^ agrees suffidentiy with tfie
contents of Euclid's commom naiwms as reduced to five te tiie most
recent text (not omitting the fourtii, that ^things whidk oolndd^ are I
equal to one another"). As rc^rds the pasimlaies^ it must be borne f
in mind that Aristotle says elsewhere* that, ''other things beinff eqQal»
that proof is the better whidi proceeds from the fewer postulates or
hypotheses or propositions.** If then we say that a geometer must
lay down as principles, first certain axicMns or common notions^ and
then an irreducible mimm$tm of postulates In tfie Aristolelkui saa«e
concerned only with the subject-matter of geometry, we are n6t hr
from describing what Euclid in fiurt does. Ais regards the pos^datM
we may imagine him sayii^: ^ Besides the common nottons there are
a few other things whidi 1 must a^ume without proof, but idiieh
differ from the common notioins in that th^ are not sdf«evident
The learner may or may not be disposed to agree to them ; but he
must accept them at the outset on the superior authority of his
teacher, and must be left to convince himself of their truth in- the
course of the investigation which follows. In the first place certain
simple constructions, the drawing and producing of a straight line,
and the drawing of a circle, must be assumed to be possible, and with
the constructions the existence of such things as straight lines and
circles ; and besides this we must lay down some postulate to form
the basis of the theory of parallels." It is true that the admission of
the 4th postulate that all right angles are equal still presents a
difficulty to which we shall have to recur.

There is of course no foundation for the idea, which has found
its way into many text-books, that *' the object of the postulates is to
declare that the only instruments the use of which is permitted in
geometry are the ruU and compassK"

§ 4. THEOREMS AND PROBLEMS.

"Again the deductions from the first principles," says Proclus, .
"are divided into problems and theorems, the former embracing the (

* Proclus, pp. 18a, 91 — 183, 13.

' Cf. Lardner's Endid : also Todhnnter.

• Atta/. poit. I. a5, 86 a 33—35.

CH. ix.$4] THEOREMS AND PROBLEMS 125

generation, division, subtraction or addition of figures, and generally
the changes which are brought about in them, the latter exhibiting
the essential attributes of each*."

" Now, of the ancients, some, like Speusippus and Amphinomus,
thought proper to call them all theorems, regarding the name of
theorems as more appropriate than that of problems to theoretic
sciences, especially as these deal with eternal objects. For there is
no becoming in things eternal, so that neither could the problem
have any place with them, since it promises the generation and
making of what has not before existed, e.g. the construction of an
equilateral triangle, or the describing of a square on a given straight
line, or the placing of a straight line at a given point. Hence they
say it is better to assert that all (propositions) are of the same kind,
and that we r^ard the generation that takes place in them as
referring not to actual making but to knowledge, when we treat things
existing eternally as if they were subject to becoming: in other words,
we may say that everything is treated by way of theorem and not
by way of problem* l^aina Oeo^pfffiarncw aXX* ov wpofiXfffiarucih

•* Others on the contrary, like the mathematicians of the school
of Menaechmus, thought it right to call them all problems, describing
their purpose as twofold, namely in some cases to furnish {iropi-
aaaOai) the thing sought, in others to take a determinate object
and see either what it is, or of what nature, or what is its property,
or in what relations it stands to something else.

''In reality both assertions are correct Speusippus is right
because the problems of geometry are not like those of mechanics,
the latter being matters of sense and exhibiting becoming and change
of every sort. The school of Menaechmus are right also because the
discoveries even of theorems do not arise without an issuing-forth
into matter, by which I mean intelligible matter. Thus forms going
out into matter and giving it shape may fairly be said to be like
processes of becoming. For we say that the motion of our thought
and the throwing-out of the forms in it is what produces the figures
in the imagination and the conditions subsisting in them. It is in
the imagination that constructions, divisions, placings, applications,
additions and subtractions (take place), but everything in the mind is
fixed and immune from becoming and from every sort of change'."
/ " Now those who distinguish the theorem from the problem say
that every problem implies the possibility, not only of that which is
predicated of its subject-matter, but also of its opposite, whereas
every theorem implies the possibility of the thing predicated but not
of its opposite as well. By the subject-matter I mean the genus
which is the subject of inquiry, for example, a triangle or a square
or a circle, and by the property predicated the essential attribute,
as equality, section, position, and the like, v When then any one

* Prodas, p. 77, 7— n. • ML pp. 77, 15—78, 8.

» Md. pp. 78, 8—79, a.

tt6 INTRODUCTION - t^^«;|4

enunciates thus, To inscrtbt an equilateral triangle in a circle, he states
a problem \ for it is also possible to inscribe in it a triangle which
is not equilateral Again, if we take the enunciation On a given
limited straight line to construct an equilateral triangle^ this is a
problem ; for it is possible also to construct one which is not equi-
lateral. But, when any one enunciates that In isosceles triangles the
angles at the base are equal, we must say that he enunciates a theorem ;
for it is not also possible that the angles at the base of isosceles
triangles should be unequal. It follows that, if any one were to use
the form of a problem and say In a semkircle to describe a right angie^
he would be set down as no geometer For every angle in a semi*
circle is rights"

" Zenodotus, who belonged to the succession of Oenopides, but
was a disciple of Andron^ distinguished the theorem from the problem
by the fact that the theorem inquires what is the property predicated
of the subject-matter in it, but the problem what is the cause of what
effect {rivo^ ivro^ ri ianv}. Hence too Fosidonius defined the one
(the problem) as a proposition in which it is inquired whether a thing
exists or not (ct ctniv ^ fuj), the other (the theorem') as a proposition
in which it is inquired what (a thing) is or of what nature (ri iarw ^
woUv Ti) ; and he said that the theoretic proposition must be put in a
declaratory form, eg., Any iHangk has two sules (together) greater than
the remaining side and /n any isosceles triangle the angles at the base
are equals but that we should state the problematic proposition as if
inquiring whether it is possible to construct an equilateral triangle
upon such and such a straight line. For there is a difference between
inquiring absolutely and indeterminately {a'nXm tc xal dopiarm^)
whether there exists a straight line from such and such a point at
right angles to such and such a straight line and investigating which
is the straight line at right angles V

** That there is a certain difference between the problem and the
theorem is clear from what has been said ; and that the Elements of
Euclid contain partly problems and partly theorems will be made
manifest by the individual propositions, where Euclid himself adds at
the end of what is proved in them, in some cases^ 'that which it was
required to do/ and in others, ' that which it was required to prove/
the latter expression being regarded as characteristic of theorems, in
spite of the fact that, as we have said, demonstration is found in
problems also. In problems, however, even the demonstration is for
the purpose of (confirming) the construction : for we bring in the
demonstration in order to show that what was enjoined has been
done ; whereas in theorems the demonstration is worthy of study for
its own sake as being capable of putting before us the nature of the
thing sought. And you will find that Euclid sometimes interweaves
theorems with problems and employs them in turn, as in the first i

^ Prodns, pp.79, II— 80,5- f

* In the text we have t6 di wftfthiiuL answering torhith withoot snbstantive : wphfikmuk .
was obviously inserted in enor.

• Froclua, pp. 80. 15—81, 4.

CH. IX. §4] THEOREMS AND PROBLEMS 127

book, while at other times he makes one or other preponderate.
For the fourth book consists wholly of problems, and the fifth of
theorems V*

Again, in his note on Eucl. I. 4, Proclus says that Carpus, the
writer on mechanics, raised the question of theorems and problems in
his treatise on astronomy. Carpus, we are told, *' says that the class
of problems is in order prior to theorems. For the subjects, the
properties of which are sought, are discovered by means of problems.
Moreover in a problem the enunciation is simple and requires no
skilled intelligence; it orders you plainly to do such and such a
thing, to construct an equilateral triangle, or, given two straight lines, to
cut off from the greater (a straight line) equal to the lesser, and what is
there obscure or elaborate in these things ? But the enunciation of a
theorem Ls a matter of labour and requires much exactness and
scientific judgment in order that it may not turn out to exceed or
fall short of the truth ; an example is found even in this proposition
(I. 4), the first of the theorems. Again, in the case of problems, one
general way has been discovered, that of analysis, by following which
we can always hope to succeed ; it is this method by which the more
obscure problems are investigated. But, in the case of theorems, the
method of setting about them is hard to get hold of since ' up to our
time,' says Carpus, * no one has been able to hand down a general
method for their discovery. Hence, by reason of their easiness, the
class of problems would naturally be more simple.' After these
distinctions, he proceeds: 'Hence it is that in the Elements too
problems precede theorems, and the Elements begin from them ; the
first theorem is fourth in order, not because the fifth* is proved from
the problems, but because, even if it needs for its demonstration none
of the propositions which precede it, it was necessary that they should
be first because they are problems, while it is a theorem. In fact, in
this theorem he uses the common notions exclusively, and in some
sort takes the same triangle placed in different positions; the
coincidence and the equality proved thereby depend entirely upon
sensible and distinct apprehension. Nevertheless, though the demon-
stration of the first theorem is of this character, the problems properly
preceded it, because in general problems are allotted the order of
precedence'.'"

Proclus himself explains the position of Prop. 4 after Props, i — 3
as due to the fact that a theorem about the essential properties of
triangles ought not to be introduced before we know that such a
thing as a triangle can be constructed, nor a theorem about the
equality of sides or straight lines until we have shown, by constructing
them, that there can be two straight lines which are equal to one
another^ It is plausible enough to argue in this way that Props. 2
and 3 at all events should precede Prop. 4. And Prop, i is used in

* Proclus, p. 81, 5 — M.

* rh rifiwTow, This should apparently be the fourth because in the next words it is
implied that none of the first three propositions are required in prorii^ it.

* Proclus, pp. 141, 19 — 143, II. * ibid, pp. 333, ai— 134, 0.

138 INTRODUCTION [CB. n. f 4

Prop. 2, and must therefore precede it But Prop, i showing how to
construct an equilateral triangle on a given base is not important, in
relation to Prop. 4, as dealing mth the ** production of triangles " in
general : for it is of no use to say, as Proclus does, that the construc-
tion of the equilateral triangle is "common to the three species (of
triangles)\'' as we are not in a position to know this at such an early
stage. The existence of triangles in general was doubtless assumed as
following from the existence of straight lines and points in one plane
and from the possibility of drawing a straight line from one point to
another.

Proclus does not however seem to reject definitely the view of
Carpus, for he goes on* : ** And perhaps problems are in order before
theorenis, and especially for those who need to ascend from the arts
which are concerned with things of sense to theoretical investigation.
But in dignity theorems are prior to problems.... It is then focuish to
blame Geminus for saying that the theorem is more perfect than the
problem. For Carpus himself gave the priority to problems in respect
of order^ and Geminus to theorems in point of more perfect eSgrnty^
so that there was no real inconsistency between the twa

Problems were classified according to the number of their possible
solutions. Amphinomus said that those which had a unique solution
(uorax»?) were called ''ordered'' (the word has dropped out in
Proclus, but it must be rvt9tiyA»a^ in contrast to the tfiird kind»
iraicr€L)\ those which had a ddfinite number of solutions "inter-
mediate " (jkkfra) ; and those with an infinite variety of solutions " un-
ordered" (£Taicra)^ Proclus gives as an example of the last tihe
problem To divide a given straight line into three farts in continued
proportion^. This is the same thing as solving the equations X'\'y-\'Z^a^
xz «>*. Proclus' remarks upon the problem show that it was solved,
like all quadratic equations, by the method of '* application of areas."
The straight line a was first divided into any two parts, (x-Vz) and j^,
subject to the sole limitation that {x-Vz) must not be less than 2y^
which limitation is the SiopurfjLo^, or condition of possibility. Then
an area was applied to (;r+ir), or'(a~'y\ ^'falling short by a square
figure" (fiCKelirov elSei rerpaywv^) and equal to the square on y. This
determines x and z separately m terms of a and y. For, if ir be the
side of the square by whiph the area (i.e. rectangle) " falls short," we
have {(a --y) -z]z ->•, whence 2z « (a -y) ± •J [{a -yY - 4j/^}. And
y may be chosen arbitrarily, provided that it is not greater than a/3.
Hence there are an infinite number of solutions. U y^aji, then, as
Proclus remarks, the three parts are equal.

Other distinctions between different kinds of problems are added
by Proclus. The word " problem," he says, is used in several senses.
In its widest sense it may mean anything " propounded " (^/Kn-ciyd-
fuvov\ whether for the purpose of instruction (jLoBrjatm) or construc-
tion (^oii}atfa>9). (In this sense, therefore, it would include a theorem.)

^ Prodns, p. 934, 91. * ibU. p. 143, 19—95.

* ihid, p. 990^ 7 — 19. ^ ibid. pp. 990^ 16—991, 6.

I

CH. ix.§4] THEOREMS AND PROBLEMS 139

But its special sense in mathematics is that of something *' propounded
with a view to a theoretic construction'."

Again you may apply the term (in this restricted sense) even to
something which is impossible, although it is more appropriately used
of what is possible and neither asks too much nor contains too little in
the shape of data. According as a problem has one or other of these
defects respectively, it is called (i) a problem in excess (irXeovd^op) or
(2) a deficient problem (jKKvirh irpopkruia). The problem in excess
(i) is of two kinds, {a) a problem in which the properties of the
figure to be found are either inconsistent {aavfifiara) or non-existent
(avvTrap/era), in which case the problem is called impossible, or (b) a
problem in which the enunciation is merely redundant : an example
of this would be a problem requiring us to construct an equilateral
triangle with its vertical angle equal to two-thirds of a right angle ;
such a problem is possible and is called "more than a problem" (fiei^op
^ irpofikfffia). The deficient problem (2) is similarly called " less than
a problem " (IXaar.aop fj wp6fi\fffia), its characteristic being that
something has to be added to the enunciation in order to convert it
from indeterminateness (aopurria) to order (rof w) and scientific deter-
minateness (Spo^ hriarrffiopuco^) : such would be a problem bidding
you " to construct an isosceles triangle," for the varieties of isosceles
triangles are unlimited. Such ''problems" are not problems ki the
proper sense {Kvplw^ Xeyofiepa wpol3Xf}fiaTa), but only equivocally".

§ 5. THE FORMAL DIVISIONS OF A PROPOSITION.

"Every problem," says Proclus', "and every theorem which is
complete with all its parts perfect purports to contain in itself all of
the following elements: enunciation (wporaai^), setting-out (licOeai^),
definition or specification {SiopuTfi6^\ construction or machinery
(tearaa-icevi^), proof (dwo&eifi^), conclusion (avfiwipao'fia). Now of
these the enunciation states what is given and what is that which is
sought, the perfect enunciation consisting of both these parts. The
setting-out marks off what is given, by itself, and adapts it before-
hand for use in the investigation. The definition or specification
states separately and makes clear what the particular thing is which
is sought. The construction ox machinery adds what is wanting to the
datum for the purpose of finding what is sought. The pro^ draws
the required inference by reasoning scientifically from acknowledged
facts. The conclusion reverts again to the enunciation, confirming
what has been demonstrated. These are all the parts of problems
and theorems, but the most essential and those which are found in all
, are enunciation, proof, conclusion. For it is equally necessary to know
beforehand what is sought, to prove this by means of the intermediate
steps, and to state the proved fact as a conclusion ; it is impossible
to dispense with any of these three things. The remaining parts
are often brought in, but are often left out as serving no purpose.

^ Proclus, p. an, 7 — 11. ■ ihid, pp. an, 13 — aaa, 14.

• ibid. pp. ao3, 1—104, 13 ; 104, a3— «05» 8.

O

H. E.

130 INTRODUCTION [ch. «• 1 5

Thus there is neither settiftg-ami nor defittUum in the problem of
constructing an isosceles triangle having each of the angles at the
base double of the remaining angle, and in most theorems there
is no construction because the setting-out suffices without any addition
for proving the required property from the data. When then do
we say that the setting-cut is wanting ? The answer is, when there
is nothing given in the enunciation \ for, though the enunciation is
in general divided into what is given and what is sought, this
is not always the case, but sometimes it states only what b sought,
i.e. what must be known or found, as in the case of the problem
just mentioned. That problem does not, in (act, state befordumd |
with what datum we are to construct the isosceles triangle having j
each of the equal angles double of the remaining angle, but (simply) |
that we are to find such a triangle.... When, then, the enuncia-
tion contains both (what is given and what is sought), in that case
we find both definition and setting-out^ but, whenever the datum
is wanting, they too are wanting. For not only is the settit^-out
concerned with the datum, but so is the definition also^ as, in the
absence of the datum, the definition will be identical with the
enunciation. In fact, what could you say in defining the object of
the aforesaid problem except that it is required to find an isosceles
triangle of the kind referred to? But that is what the enundaiioH
stated. If then the enunciation does not include, on the one hand,
what is given and, on the other, what is sought, there is no setting-out
in virtue of there being no datum, and the definition is left out in
order to avoid a mere repetition of the enumiation!^

The constituent parts of an Euclidean proposition will be readily
identified by means of the above description. As regards the defi-
nition or specification (Biopicfio^) it is to be observed that we have
here only one of its uses. Here it means a closer definition or descrip-
tion of the object aimed at, by means of the concrete lines or figures
set out in the exOeai^ instead of the general terms used in the enun-
ciation ; and its purpose is to rivet the attention better, as Proclus
indicates in a later passage {rpoirov rwh irpoaex^la^ iarlv alrio^ o
htopiaiUs^y.

The other technical use of the word to signify the limitations to
which the possible solutions of a problem are subject is also described
by Proclus, who speaks of Biopia^ioi determining ''whether what is
sought is impossible or possible, and how far it is practicable and in
how many ways'" ; and the Biopurfio^ in this sense appears in Euclid
as well as in Archimedes and Apollonius. Thus we have in Eucl. I.
22 the enunciation *'From three straight lines which are equal to
three given straight lines to construct a triangle," followed imme- «
diately by the limiting condition (Bioptafi6<;). "Thus two of the
straight lines taken together in any manner must be greater than the
remaining one." Similarly in VI. 28 the enunciation "To a given
straight line to apply a parallelogram equal to a given rectilineal

> Proclus, p. 108, II. * Mii. p. 303, 3.

CH. IX. is] FORMAL DIVISIONS OF A PROPOSITION 131

figure and falling short by a parallelogrrammic figure similar to a
given one " is at once followed by the necessary condition of possi-
bility: "Thus the given rectilineal figure must not be greater than
that described on half the line and similar to the defect."

Tannery supposed that, in giving the other description of the
hiopwiU^ as quoted above, Proclus, or rather his g^ide, was using the
term incorrectly. The SiopiafAo^ in tlie better known sense of the
determination of limits or conditions of possibility was, we are told,
invented by Leon. Pappus uses the word in this sense only. The
other use of the term might. Tannery thought, be due to a confusion
occasioned by the use of the same words {Set £17) in introducing the
parts of a proposition corresponding to the two meanings of the word
Siopur/AOf;^ On the other hand it is to be observed that Eutocius
distinguishes clearly between the two uses and implies that the differ-
ence was well known*. The Biopiafiif: in the sense of condition of
possibility follows immediately on the enunciation, is even part of it ;
the Biopiafio^ in the other sense of course comes immediately after the

Proclus has a useful observation respecting the concltision of a
proposition'. "The conclusion they are accustomed to make double
in a certain way : I mean, by proving it in the given case and then
drawing a general inference, passing, that is, from the partial con-
clusion to the general. For, inasmuch as they do not make use of
the individuality of the subjects taken, but only draw an angle or a
straight line with a' view to placing the datum before our eyes, they
consider that this same fact which is established in the case of the
particular figure constitutes a conclusion true of every other figure of
the same kind. They pass accordingly to the general in order that
we may not conceive the conclusion to be partial. And they are
justified in so passing, since they use for the demonstration the par-
ticular things set out, not qud particulars, but qud typical of the rest
For it is not in virtue of such and such a size attaching to the angle
which is set out that I effect the bisection of it, but in virtue of its
being rectilineal and nothing more. Such and such size is peculiar to
the angle set out, but its quality of being rectilineal is common to all
rectilineal angles. Suppose, for example, that the given angle is a
right angle. . If then I had employed in the proof the fact of its being
right, I should not have been able to pass to every species of recti-
lineal angle ; but, if I make no use of its being right, and only consider
it as rectilineal, the argument will equally apply to rectilineal angles
in general."

' La Ciomitrie grecque^ p. 149 note. Where det 9^ introduces the closer description of
the problem we may translate, "it is then required** or '*thus it is required" (to construct etc):
when it introduces the condition of possibility we may translate **thus it is necessary etc.*'
Heiberg originally wrote dct M in the latter sense in I. 13 on the authority of Produs and
Eutocius, and against that of the Mss. Later, on the occasion of xi. 33, he observed that he
should have followed the mss. and written dct ^ which he found to be, after all, the right
reading in Eutocius (Apollonius, ed. Heiberg, 11. p. 178). det ^ is also the expression usied
by Diophantus for introducing conditions of possibility.

' See the passage of Eutocius referred to in last note. ' Proclus, p. 107, 4 — 15.

9—2

132 INTRODUCTION [cb. dl |6

§ 6. OTHER TECHNICAL TERMS.

I. Things said to be given.

Proclus attaches to his description of the formal ^ivisiona of a
proposition an explanation of the different senses in which the word
given or datum (BeSofiit^ov) is used in geometry. ''Everything that is
given is given in one or other of the following ways^ in fosiiiam^ in
ratio, in magnitude, or in specks. The point is given in pasiium only,
but a line and the rest may be given in all the senses^"

The illustrations which Proclus gives of the four senses in which a
thing may be given are not altogether happy, and, as regards things
which are given in position^ in fn^gnitude^ and in species, it is best, I
think, to follow the definitions given by Euclid himself in his book of
Data. Euclid does not mention the fourth class, things given in ratio^
nor apparently do any of the great eeometers.

( 1 ) Given in position really needs no definition ; and, when Euclid
says (Data, Def 4) that ^Points, lines and angles are said to ht given
in position which always occupy the same place,| we are not really
the wiser.

(2) [Given in magnitude is defined thus {Data^ De£ l): ''Aieas,
lines and angles are called given in tnagnitude to which we can find
equals."! Proclus' illustration is in this case the following: when, he
says, two unequal straight lines are given from the greater of which
we have to cut ofT a straight line equal to the lesser, & straight lines
are obviously given in magnitude^ *' for greater and less, and finite
and infinite are predications peculiar to magnitude." But he does not
explain that part of the implication of the term is that a thing is given
in magnitude only, and that, for example, its position is not given and
is ajnatter of indifference.

j(3) Given in species. Euclid's definition {Data, Def. 3) is:
'' Rectilineal figures are said to b^ given in species in which the angles
are severally given and the ratios of the sides to one another are

given/' I And this is the recognised use of the term (cf. Pappus,

passim). Proclus uses the term in a much wider sense for which I am
not aware of any authority. Thus, he says, when we speak of (bisect-
ing) a given rectilineal angle, the angle is given in species by the word
rectilineal, which prevents our attempting, by the same method, to
bisect a curvilineal angle ! On Eucl. i. 9, to which he here refers, he
says that an angle is given in species when e.g. we say that it is right
or acute or obtuse or rectilineal or " mixed," but that the actual angle
in the proposition is given in species only. As a matter of fact, we
should say that the actual angle in the figure of the proposition is
given in magnitude and not in species, part of the implication of given
in species being that the actual magnitude of the thing given in species
is indifferent ; an angle cannot be given in species in this sense at a^.
The confusion in Proclus' mind is shown when, after saying that a
right angle is given in species, he describes a third of a right angle as
given in magnitude.

' Prodiu, p. «05, 13-^15.

CH. ix.§6] OTHER TECHNICAL TERMS 133

No better example of what is meant by given in species^ in its
proper sense, as limited to rectilineal figures, can be quoted than the
g^ven parallelogram in Eucl. VI. 28, to which the required parallelo-
gram has to he made similar; the former parallelogram is in fact
given in species^ though its actual size, or scale, is indifferent

(4) I Given in ratio presumably means something which is given
by means of its ratio to some other given thing.( This we gather from
Proclus' remark (in his note on I. 9) that an angle may be given in
ratio ** as when we say that it is double and treble of such and such an
angle or, generally, greater and less." The term, however, appears to
have no authority and to serve no purpose. Proclus may have
derived it from such expressions as "in a given ratio" which are
common enough.

2. Lemma.

**The term Umma" says Proclus*, "is often used of any proposition
which is assumed for the construction of something else : thus it is a
common remark that a proof has been made out of such and such
lemmas. But the special meaning of lemma in geometry is a
proposition requiring confirmation. For when, in either construction
or demonstration, we assume anything which has not been proved but
requires argument, then, because we regard what has been assumed as
doubtful in itself and therefore worthy of investigation, we call it a
lemma\ differing as it does from the postulate and the axiom in being
matter of demonstration, whereas they are immediately taken for
granted, without demonstration, for the purpose of confirming other
things. Now in the discovery of lemmas the best aid is a mental
aptitude for it. For we may see many who are quick at solutions and
yet do not work by method ; thus Cratistus in our time was able to
obtain the required result from first principles, and those the fewest
possible, but it was his natural gift which helped him to the discovery.

* Proclus, pp. ail, i — an, 4.

* It would appear, sajrs Tannery (p. 15111.), that Geminus understood a lemma as being
simply XcMi/Sar^eror, something assumed (cf. the passage of Proclus, p. 73, 4, relating to
Menaechmus' view of eiements) : hence we cannot consider ourselves authorised in attributing
to Geminus the more technical definition of the term here given by Proclus, according to
which it b only used of propositions not proved beforehand. This view of a lemma must
be considered as relatively modem. It seems to have had its origin in an imperfection of
method. In the course of a demonstration it was necessary to assume a proposition which
required proof, but the proof of which would, if inserted in the particular place, break the
thread ot the demonstration : hence it was necessary either to prove it beforehand as a
preliminary proposition or to postpone it to be proved afterwards {in i^ 5ctx^cra«).
When, after the time of Geminus, the progress of oneinal discovery in geometry was arrested,
geometers occupied themselves with the study and elucidation of the works of the great
mathematicians who had preceded them. This involved the investigation of propositions
explicitly quoted or tsicitly assumed in the great classical treatises; and naturally it was found
that several such remained to be demonstrated, either because the authors had omitted
them as being easy enough to be left to the reader himself to prove, or because books in
irhich they were proved had been lost in the meantime. Hence arose a class of complementary
or auxiliary propositions which were called iemptas. Thus Pappus gives in his Book vn a
collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the
so-called "Treasury of Analysis " (r6rot dvaXu^erot). When Proclus goes on to distinguish
three methods of discovering lemmas, analysis^ dnnsicn, and reductic ad adsurditmf he seems
to imply that the principal business of contemporary geometers was the investigation of these
auxiliary propositions.

Iu~

134 INTRODUCTION [CB. n. |6

Nevertheless certain methods have been handed down. The finest b
the method which by means of attafysis carries the thing aought up to
an acknowledged principle, a method which Plato, as tney say. ccMn-
municated to Leodamas\ and by which the latter, too, is flAid to have
discovered many things in geometry. The second is tiie method of
division^ which divides into its parts the genus proposed for con-
sideration and gives a starting-point for the demonstration b^ means
of the elimination of the other elements in the construction of what is
proposed, which method also Plato extolled as being of assistance to
all sciences. The third is that by means of the ndMctw ad aisunbim^
which does not show what is sought directly, but refutes its opposite
and discovers the truth incidentaUy.**

3. Case.

"^ The case^ (wrScif:)" Proclus proceeds^ ''announces difierent ways
of construction and alteration <^ positions due to the transposition of
points or lines or planes or solidk And, in general, all its varieties
are seen in the figure, and this is why it b called our, being a trans-
position in the construction."

4. Porism.

** The term porism is used also <^ certain problems such as the
Porisms written by Euclid. But it is specially used when from what
has been demonstrated some otiher theorem b revealed at the same
time without our propounding it which theorem has on thb very
account been called a porism (corollary) as being a sort of inddenttd
gain arising from the scientific demonstration*." Cf the note on L 15.

* This punge and another from Dioeenes Laertias (ill. 34, p. 7^ ed. Cobet) to the cflect
that *' He [Plato] explained (tlnrrh^^r^ to Lcodamas of Thasos the method of inquiry by
analysis " have been commonly understood as ascribing to Plato the invention of the method
of analvsts ; but Tannery points out forcibly (pp. 1 ta, 113) how difficult it is to explain in
what Plato's discovery could have consisted if tmalfsis be taken in the sense attributed to it
In Pappus, where we can see no more than a series of successive reductions of a problem
until It is finally reduced to a known problem. On the other hand, Proclus' words about •
carrying up the thing sought to *' an acknowledged principle " suggest that what he had in

mind was the process described at the end of Book vi of the Republic by which the dialec<
tidan (unlike tne mathematician) uses hypotheses as stepping-stones np to a principle which
is not hypothetical, and then is able to descend step Dy step verifying every one of the
hypotheses by which he ascended. This description does not of course rder to mathematical
analvsis, but it may have given rise to the ioea that analysis was Plato's discovery, since
anafytis and synthesis following each other are related in the same way as the upward and
the downward progression in the dialectician's intdlectual method. And it may be that
Plato's achievement was to observe the importance, from the pomt of view of logiod rigour,
of the confirmatory synthesis following analysis, and to regularise in this way and elevate
Into a completely irrefragable method the partial and uncertain analysis upon which the
works of his predecessors depended.

* Here apin the successive bipartitions of genera into species such as we find in the
Sophist and Republic have very little to say to geometry, and the very fact that they are hoe
mentioned side bv side with analysis suggests that Proclus confused the latter with the
philosophical method of Ref. vi.

* Tannery rightly remarks (p. 15a) that the subdivision of a theorem or problem Into
several cases is foreign to the really classic form ; the ancients preferred, where necessary, to
multiply enunciations. As, however, some omissions necessarily occurred, the writers of
lemmas naturally added separate casa, which in some instances found their way into die text.
A good example is Eudid i. 7, the second case of which, as it appears in our text-books,
was interpolated. On the commentary of Proclus on this proposition Th. Taylor rightly
remarks that *' Euclid everywhere avokfs a multitude of cases."

* Produs, p. SIS, 5— II.

* Tannery notes however that, so fin'^lrom distingutshing his corollaries from the con-

CH. ix.§6] OTHER TECHNICAL TERMS 135

5. Objection.

" The objectuni (Ivaraa-i^) obstructs the whole course of the argu-
ment by appearing as an obstacle (or crying ' halt/ ehravTwaa) either
to the construction or to the demonstration. There is this difference
between the objection and the case, that, whereas he who propounds
the case has to prove the proposition to be true of it, he who makes
the objection does not need to prove anything : on the contrary it is
necessary to destroy the objection and to show that its author is
saying what is false*."

That is, in general the objection endeavours to make it appear that
the demonstration is not true in every case ; and it is then necessary
to prove, in refutation of the objection, either that the supposed case
is impossible, or that the demonstration is true even for that case. A
good instance is afforded by Eucl. I. 7. The text- books give a second
case which is not in the original text of Euclid. Proclus remarks on
the proposition as given by Euclid that the objection may conceivably
be raised that what Euclid declares to be impossible may after all be
possible in the event of one pair of straight lines falling completely
within the other pair. Proclus then refutes the objection by proving
the impossibility in that case also. His proof then came to ht given
in the text-books as part of Euclid's proposition.

The objection is one of the technical terms in Aristotle's logic and
its nature is explained in the Prior Analytics^ "An objectiofi is a
proposition contrary to a proposition.... Objections are of two sorts,
general or partial.... For when it is maintained that an attribute
belongs to every (member of a class), we object either that it belongs
to none (of the class) or that there is some one (member of the class)
to which it does not belong."

6. Reduction.

This is again an Aristotelian term, explained in the Prior
Analytics^ It is well described by Proclus in the following passage :

" Reduction {dwayaryi]) is a transition from one problem or theorem
to another, the solution or proof of which makes that which is pro-
pounded manifest also. For example, after the doubling of the cube
had been investigated, they transformed the investigation into another
upon which it follows, namely the finding of the two means ; and from
that time forward they inquired how between two given straight lines
two mean proportionals could be discovered. And they say that the
first to effect the reduction of difficult constructions was Hippocrates of
Chios, who also squared a lune and discovered many other things in
geometry, being second to none in ingenuity as r^ards constructions*."

dnsioDS of his propositions, Euclid inserts them before the closing words "(being) what it
was required to do*' or "to prove.** In fact the porism-corollary is with Euclid rather a
modifira form of the regular conclusion than a separate proposition.

* Proclus, p. 211, 18 — 13.
■ ^na/. /nor. ii. 26, 69 a 37.

' idid, II. 35, 69 a 10.

* Proclus, pp. 113, 14—313, II. This passage has frequently been taken as crediting
Hippocrates with the discovery of the method of geometrical reduction : cL Taylor (Transla-
tion of Proclus, II. p. 16), Allman (p. 41 n., 59), Gow (pp. 169, 170). As Tannery remarks
(p. 110), if the particular reduction of the duplication problem to that of the two means is

13d INTROftUCtlOK {^ hu ii

7, Reductio ad absurdum.

This is variously called by Aristotle "rediuiia ad absurdum^' (^ m
nJ m^aTov awaywy ri)\ *" proof /^r impossibik'^ {i} Zia rov aSvudrau
Bei^i^i or airoSfif*?)*, "proof leading to the impossible" {^ ca9 to
tiBvparoif ayov^a fhraBeih^y. It is part of *' proof (starting) from a
hypothesis**' (cf vwo0€iJ€oi^). "All (syllogisms) which reach the
conclusion fier imfossibili reason out a conclusion which is false, and
they prove the original contention (by the method starting) from a
hypothesis, when something impossible results from assuming the
contradictory of the original contention, as, for example, when it is
proved that the diagonal (of a square) is incommensurable because,
if it be assumed commensurable, it will follow that odd (numbers)
are equal to even (numbers)*/* Or again, *' proof (leading) to the
impossible differs from the direct (S€*<t*«^5) in that it assumes what
it desires to destroy [namely the hypothesis of the falsity of the
conclusion] and then reduces it to something admittedly false, whereas
the direct proof starts from premisses admittedly true'.'*

Proclus has the following description of the reductw ad absurdum.
'* Proofs by reductio ad ahsurdum \n every case reach a conclusion
manifestly impossible, a conclusion the contradictory of which is
admitted. In some cases the conclusions are found to conflict with
the common notions, or the postulates, or the hypotheses (from which
we started) ; in others they contradict propositions previously estab-
lished^".*/' Every reducth ad absurdum assumes what conflicts with
the desired result, then, using that as a basis, proceeds until it arrives
at an admitted absurdity, and^ by thus destroying the hypothesis,
establishes the result originally desired. For it is necessary to under-
stand generally that all mathematical arguments either proceed from
the first principles or lead back to them, as Porphyry somewhere says.
And those which proceed from the hrsX principles are again of two
kinds, for they start cither from common notions and the clearness of
the self-evident alond or from results previously proved \ while those
which lead back to the principles are either by way of assuming the
principles or by way of destroying them. Those which assume the
principles are called analyses^ and the opposite of these are sjnfAeses —
for it is possible to start from the said principles and to proceed in
the regular order to the desired conclusion, and this process is syn-
ihesis — while the arguments which would destroy the principles are

the first noted in hisioryj it « difficult to £tipf>o*ic that it was really the firet ; for Hippocrates
x&u&l have found instances of ii in the Pythi^ortjji geomctiy- lirelschneidei, I ihiiiL, tonics
nearer the truth when he boldly (p. 99) translates: "Tms reduction c/tie t/artsaiii ram-
struct ion is said to have been fint given by Hippocrates.'* The words are wfitrntf Ik fun
xAr Aropavfidwm BtaypofiMrif rV lva7«ryV voci^o^^t which must, literally, be translated
as in the text above; but, when Proclus speaks vaguely of "difficult oonstractioiis,'' he
probably means to say simply that '* this firrt recorded instance of a reduction of a difficult
construction is attributed to Hippocrates."'

I Aristotle, Anai.frwr, i. 7t ^9 b 5 ; i. 44, 50 a 30.

* Uid. I. 91, 39 b 51 ; I. ap, 45 a 55.

* Anai, past. I. 94, 85 a 10 etc. ^ Anal, prior, i. 93, 40 b 95.

* Anal. prUr. I. 93, 41 a 94. * iHd. 11. 14, 69 b 99.
^ Produs, p. 954, 99—97.

r

CH. IX.46] OTHER TECHNICAL TERMS 137

called reductiones ad absurdum. For it is the function of this method
to upset something admitted as clear*."

8. Analysis and Synthesis.

It will be seen from the note on Eucl. XIII. i that the MSS. of the
Elements contain definitions of Analysis and Synthesis followed by
alternative proofs of xill. i — 5 after that method. The definitions and
alternative proofs are interpolated, but they have great historical
interest because of the possibility that they represent an ancient
method of dealing with these propositions, anterior to Euclid. The
propositions give properties of a line cut "in extreme and mean ratio,"
and they are preliminary to the construction and comparison of the
five regular solids. Now Pappus, in the section of his CoUection dealing
with the latter subject*, says that he will give the comparisons between
the five figures, the pyramid, cube, octahedron, dodecahedron and
icosahedron, which have equal surfaces, " not by means of the so-called
analytical vdK^VTf, by which some of the ancients worked out the proofs,
but by the synthetical method*...." The conjecture of Bretschneider
that the matter interpolated in Eucl. xill. is a survival of investiga-
tions due to Eudoxus has at first sight much to commend it^ In the
first place, we are told by Proclus that Eudoxus " greatly added to
the number of the theorems which Plato originated regarding the
section^ and employed in them the method of analysis'.** It is obvious
that " the section " was some particular section which by the time of
Plato had assumed great importance ; and the one section of which
this can safely be said is that which was called the " golden section,"
namely, the division of a straight line in extreme and mean ratio
which appears in Eucl. II. 1 1 and is therefore most probably Pytha-
gorean. Secondly, as Cantor points out*, Eudoxus was the founder
of the theory of proportions in the form in which we find it in Euclid
v., VI., and it was no doubt through meeting, in the course of his
investigations, with proportions not expressible by whole numbers
that he came to realise the necessity for a new theory of proportions
which should be applicable to incommensurable as well as commen-
surable magnitudes. The "golden section" would furnish such a case.
And it is even mentioned by Proclus in this connexion. He is
explaining' that it is only in arithmetic that all quantities bear
"rational" ratios (/J»;toc >jlrio^) to one another, while in geometry there
are '* irrational " ones (apptfro^) as well. " Theorems about sections
like those in Euclid's second Book are common to both [arithmetic
and geometry] except that in which the straight line is cut in extreme
and mean ratio^**

» Proclus, p. 355, 8— «6.

■ Pappus, V. p. 410 sqq. • ibid. pp. 410, 17 — 4H, a.

^ Bretschneider, p. 108. See however Heibeig's recent suggestion (Paralipomena tu
EukUd in Hermes^ xxxviii., 1003) that the author was Heron. The suggestion is based
on a comparison with the remarks on analysis and synthesis Quoted from Heron by an-NairizI
(ed. Cortxe, p. 89) at the beginning of his commentary on Eucl. Book il. On the whole,
this suggestion commends itself to me more than that of Bretschneider.

* Proclus, p. 67, 6. * Cantor, Gtsch, d, Maih, if, p. 941.

' Proclus, p. 60, 7 — 9. • ibid, p. 60, 16 — 19.

138 INTRODUCTION [cB.a.f6

The definitions of Analysis and Symthssis interpolated in EucL
XIII. are as follows (I adopt the reading of B and V, the only in-

telligible one, for the second^

" Analysis is an assumption of that which is sought as if it were
admitted < and the passage > through its consequences to sbmething
admitted (to be) true.

" Synthesis is an assumption of that which is admitted < and the
passage > through its consequences to the finishing or attainment of
what is sought"

The language is by no means clear and has, at the best, Id be
filled out.

Pappus has a fuller account* :

"* The so-called avakuiiktya^ (' Treasury of Analysis ') is. to put it
shortly, a special body of doctrine provided for the use of those who.
after finishing the ordinary Elements, are desirous of acquiring the
power of solving problems which may be set them involving (the
construction of) lines, and it is useful for this alone. It is the woric
of three men, Euclid the author of the Elements, ApoUonius of Perga,
and Aristaeus the elder, and proceeds by way of analysis and synthesis.

" Analysis then takes that which is sought as if it were admitted
and passes from it through its successive consequences to something
which is admitted as the result of synthesis: for in analysis we assume
that which is sought as if it were (already) done (tctoi^X ^^'^ ^"^
inquire what it is from which this results,^ and again what is the ante-
cedent cause of the latter, and so on, until by so retracing our steps
we come upon something already known or belonging to the class of
first principles, and such a method we call analysis as being solution
backwards {avdiraKiv \vcivy,

" But in synthesis, reversing the process, we take as already done
that which was last arrived at in the analysis and, by arranging in
their natural order as consequences what were before antecedents,
and successively connecting them one with another, we arrive finally
at the construction of what was sought ; and this we call synthesis.

'' Now analysis is of two kinds, the one directed to searching for
the truth and called theoretical, the other directed to finding what we
are told to find and called problematicaL (i) In the theoretical kind
we assume what is sought as if it were existent and true, after which
we pass through its successive consequences, as if they too were true
and established by virtue of our hypothesis, to something admitted :
then {a\ if that something admitted is true, that which is sought will
also be true and the proof will correspond in the reverse order to the
analysis, but {b\ if we come upon something admittedly false, that
which is sought will also be false. (2) In the problematical kind we
assume that which is propounded as if it were known, after which we
pass through its successive consequences, taking them as true, up to
something admitted : if then {a) what is admitted is possible and
obtainable, that is, what mathematicians call given^ what was originally
proposed will also be possible, and the proof will again correspond in

* PApptts, vii. pp. 634 — 6.

il

CH.ix.§6] OTHER TECHNICAL TERMS 139

reverse order to the ianalysis, but if (d) we come upon something
admittedly impossible, the problem will also be impossible."

The ancient Analysis has been made the subject of careful studies
by several writers during the last half-century, the most complete
being those of Hankel, Duhamel and Zeuthen ; others by Ofterdinger
and Cantor should also be mentioned^

The method is as follows. It is required, let us say, to prove that
a certain proposition A is true. We assume as a hypothesis that A
is true and, starting from this we find that, if A is true, a certain
other proposition B is true ; if B is true, then C ; and so on until
we arrive at a proposition K which is admittedly true. The object
of the method is to enable us to infer, in the reverse order, that, since
K is true, the proposition A originally assumed is true. Now
Aristotle had already made it clear that false hypotheses might lead
to a conclusion which is true. There is therefore a possibility of error
unless a certain precaution is taken. While, for example, B may be a
necessary consequence of A, it may happen that A is not a necessary
consequence of B. Thus, in order that the reverse inference from the
truth of K that A is true may be logically justified, it is necessary
that each step in the chain of inferences should be unconditionally
convertible. As a matter of fact, a very large number of theorems in
elementary geometry are unconditionally convertible, so that in practice
the difficulty in securing that the successive steps shall be convertible
is not so great as might be supposed. But care is always necessary.
For example, as Hankel says', a proposition may not be uncon-
ditionally convertible in the form in which it is generally quoted.
Thus the proposition " The vertices of all triangles having a common
base and constant vertical angle lie on a circle " cannot be converted
into the proposition that "All triangles with common base and vertices
lying on a circle have a constant vertical angle*'; for this is only true
if the further conditions are satisfied (i) that the circle passes through
the extremities of the common base and (2) that only that part of the
circle is taken as the locus of the vertices which lies on one side of the
base. If these conditions are added, the proposition is unconditionally
convertible. Or again, as Zeuthen remarks', K may be obtained by
a series of inferences in which A or some other proposition in the
series is only apparently used ; this would be the case e.g. when the
method of modem algebra is being employed and the expressions on
each side of the sig^ of equality have been inadvenently multiplied
by some composite magnitude which is in reality equal to zero.

Although the above extract from Pappus does not make it clear
that each step in the chain of argument must be convertible in the
case taken, he almost implies this in the second part of the definition
of Analysis where, instead of speaking of the consequences B, C...

^ Hankel, Zur Gtschichte der Mathematik in Altertkum und MUUlalter^ 1 874, pp. 137— 1 50 ;
Duhamel, Dts mStkodes dans Us sciences de raisonnement^ Part I., 3 ed., Paris, 1885, pp. 39 — 68 ;
2^then, Gesckickie der Mathematik im Altertum und Mitteiaiter^ 18961 pp. 91 — 104;
Ofterdinger, Beitrage tmr Gesckickie der grieckiscken Maikimaiik^ Ulm, 1800; Cantor,
Gesckickie der Matkematik, ij, pp. a 10 — 1.

' Hankel, p- 139. ' Zeuthen, p. 103.

I40 INTRODUCTION [CB. OL f 6

successively following from A» he suddenly changes the expreitkm
and says that we inquire wkai ii is (B)fivm which A fallows (A bdng
thus the consequence of B, instead of the reverse), and then what
(viz. C) is the antecedent cause of B; and in practice tlie Greeks
secured what was wanted by always insisting on the analysis being
confirmed by subsequent synthesis, that is, they laboriously worked
backwards the whole way from K to Ap reversing the order of the
analysis, which process would undoubtedly bring to light any flaw
which had crept into the argument through tibe accidental neglect of
the necessary precautions.

Reductio ad absurdum a variety of analysis.

In- the process of analysis starting from the hypothesis that a
proposition A is true and passing through B, C... as successive con-
sequences we may arrive at a proposition K which, instead of being
admittedly true, is either admittedly false or the contradictory of the
original hypothesis A or of some one or more of the propositions B^ C...
intermediate between A and K. Now correct inference from a true
proposition cannot lead to a false ^proposition ; and in this case there-
fore we may at once conclude, wiuout any inquiry whether the
various steps in the ailment are convertible or not, that the hypo-
thesis A is false, for, if it were true, all the consequences correctly
inferred from it would be true and no incompatibility could arise.
This method of proving that a given hypothesis is false furnishes an
indirect method of proving that a given hypodiesis A is true^ since we
have only to take the contradictory of A and to prove that it is false.
This is the method of reductio ad absurdum^ which is therefore a variety
of analysis. The contradictory of A, or not- A, will generally include
more than one case and, in order to prove its falsity, each of the cases
must be separately disposed of: e.g., if it is desired to prove that a
certain part of a figure is equal to some other part, we take separately
the hypotheses (i) that it is greater^ (2) that it is less^ and prove
that each of these hypotheses leads to a conclusion either admittedly
false or contradictory to the hypothesis itself or to some one of its
consequences.

Analysis as applied to problems.

It is in relation to problems that the ancient analysis has the
greatest significance, because it was the one general method which
3ie Greeks used for solving all "the more abstruse problems" (rii
daa^arepa' rAv TrpofiXsffLdrc^py. I

We have, let us suppose, to construct a figure satisfying a certain
set of conditions. If we are to proceed at all methodically and not
by mere guesswork, it is first necessary to "analyse" those conditions.
To enable this to be done we must get them clearly in our minds,
which is only possible by assuming all the conditions to be actually
fulfilled, in other words, by supposing the problem solved. Then we
have to transform those conditions, by all the means which practice in
such cases has taught us to employ, into other conditions which are
necessarily fulfilled if the original conditions are, and to continue this

^ Prodiu, p. 94*, 16, 17.

CH. ix.§6] OTHER TECHNICAL TERMS 141

transformation until we at length arrive at conditions which we
are in a position to satisfy ^ In other words, we must arrive at
some relation which enables us to construct a particular part of
the figure which, it is true, has been hypothetically assumed and
even drawn, but which nevertheless really requires to h^ found in
order that the problem may be solved. From that moment the
particular part of the figure becomes one of the data, and a fresh
relation has to be found which enables a fresh part of the figure
to be determined by means of the original data and the new one
together. When this is done, the second new part of the figure also
belongs to the data ; and we proceed in this way until all the parts
of the required figure are found*. The first part of the analysis
down to the point of discovery of a relation which enables
us to say that a certain new part of the figure not belonging
to the original data is given, Hankel calls the transformation ; the
second part, in which it is proved that all the remaining parts of
the figure are "given," he calls the resolution. Then follows the
synthesis^ which also consists of two parts, (i) the construction^ in
the order in which it has to be actually carried out, and in general
following the course of the second part of the analysis, the resolution ;
(2) the demonstration that the figure obtained does satisfy all the given
conditions, which follows the steps of the first part of the analysis,
the transformation, but in the reverse order. The second part of
the analysis, the resolution, would be much facilitated and shortened
by the existence of a systematic collection of Data such as Euclid's ^
book bearing that title, Jbonsisting of propositions proving that, if
in a figure certain parts or relations arc given, other parts or relations
are also given(/ As regards the first part of the analysis, the trans-
formation, the usual rule applies that every step in the chain must
be unconditionally convertible; and any failure to observe this
condition will be brought to light by the subsequent synthesis.
The second part, the resolution, can be directly turned into the
construction since that only is given which can be constructed by
the means provided in the Elements.

It would be difllicult to find a better illustration of the above than
the example chosen by Hankel from Pappus.^

Given a circle ABC and two joints D, E external to it, to draw
straight lines DB, l£.Efrom D,E to a point B on the circle such that,
if DB, IL^ produced meet the circle again in C, A, AC shcdl be parallel
to DE.

Analysis.

Suppose the problem solved and the tangent at A drawn, meeting
ED produced in F.

(Part I. Transformation.)

Then, since AC is parallel to D£, the angle at C is equal to the
angle CDE.

But, since FA is a tangent, the angle at C is equal to the angle FAE.

Therefore the angle FAE is equal to the angle CDE, whence A,
By D, F are concyclic.

* 2^then, p. 93. • Hankel, p. 141. ' » Pappus, vii. pp. 830— «.

143 INTRODUCTION

I

[CH. UL |6 I

Therefore the rectangle AE^ EB is equal to tiie rectan^e FR^
ED.

(Part II. Resolution.)

But the rectangle AE^ EB is given,
because it is equal to the square on the
tangent from E.

Therefore the rectangle FE^ ED is
given;

and, since ED is given, FE is nven (in
length). [Data, 57J

But FE is given in position also,- so
that F is also given. [Daia^ 27.]

Now FA is the tangent from a given point F to a circle ABC
given in position ;
therefore FA is given in position and magnitude. [Daia^ 9a]

And F is given ; therefore A is given.

But E is also given ; therefore the straight line AE is given in
position. [Dttia^ 26.]

And the circle ABC is given in position ;
therefore the point B is also given. [Data^ 35.]

But the points D, E are idso given ;
therefore the straight lines DB^ BE are also given in position.

Synthesis.

(Part I. Construction.)

Suppose the circle ABC and the points A E given.

Take a rectangle contained by ED and by a certain strai^t
line EF equal to the square on the tangent to the circle from E.

From F draw FA touching the circle in A ; join ABE and then
DB^ producing DB to meet the circle at C. Join A C.

1 say then that AC is parallel to DE.

(Part II. Demonstration.)

Since, by hypothesis, the rectangle FEy ED is equal to the square
on the tangent from Ey which again is equal to the rectangle AEy EB^
the rectangle AEy EB is equal to the rectangle FEy ED.

Therefore AyByDyF are concyclic,
whence the angle FAE is equal to the angle BDE.

But the angle FAE is equal to the angle ACB in the alternate
segment ;
therefore the angle A CB is equal to the angle BDE.

Therefore AC\s parallel to DE.

In cases where a Siopurfio^ is necessary, i.e. where a solution is
only possible under certain conditions, the analysis will enable those ]
conditions to be ascertained. Sometimes the Siopiafio^ is stated and
proved at the end of the analysis, e.g. in Archimedes, On the Sphere
and Cylindery II. 7 ; sometimes it is stated in that place and the proof
postponed till after the end of the synthesis, e.g. in the solution of
the problem subsidiary to On tfte Sphere and Cylindery II. 4, preserved
in Eutocius' commentary on that proposition. The analysis should
also enable us to determine the number of solutions of which the
problem is susceptible.

'

CH. ix.§7] THE DEFINITIONS 143

§ 7. THE DEFINITIONS.

General. " Real " and " Nominal " Definitions.

It is necessary, says Aristotle, whenever any one treats of any
whole subject, to divide the genus into its primary constituents, those
which are indivisible in species respectively: e.g. number must be
divided into triad and dyad ; then an attempt must be made in this
way to obtain definitions, e.g. of a straight line, of a circle, and of
a right angle'.

The word for definition is 2/)09. The original meaning of this
word seems to have been "boundary," "landmark." Then we have
it in Plato and Aristotle in the sense of standard or determining
principle ("id quo alicuius rei natura constituitur vel definitur,"
Index AristotelicusY ; and closely connected with this is the sense of
definition. Aristotle uses both 8/309 and opurfio^ for definition, the
former occurring more frequently in the Topics, the latter in the
MetapAj^sics.

Let us now first be clear as to what a definition does not do.
There is nothing in connexion with definitions which Aristotle takes
more pains to emphasise than that a definition asserts nothing as to
the existence or non-existetice ol the thing defined. It is an answer
to the question what a thing is (r/ ^cm), and does not say tliat it
is (oTi itrrC). The existence of the various things defined has to be
proved, except in the case of a few primary things in each science,
the existence of which is indemonstrable and must be assumed among
the first principles of each science ; c.g[. points and lines in geometry
must be assumed to exist, but the existence of everything else must
be proved. This is stated clearly in the long passage quoted above
under First Principles'. It is reasserted in such passages as the
following. "The (answer to the question) what is a man and the
fact that a man exists are different things*.*' " It is clear that, even
according to the view of definitions now current, those who define
things do not prove that they exist'." "We say that it is by
demonstration that we must show that everything exists, except
essence (c* ft^ ovala tlri). But the existence of a thing is never
essence; for the existent is not a genus. Therefore there must be
demonstration that a thing exists. Thus, what is meant by triangle
the geometer assumes, but that it exists he has to prove V "Anterior
knowledge of two sorts is necessary : for it is necessary to presuppose,
with regard to some things, that they exist \ in other cases it is
necessary to understand what the thing described is, and in other
cases It is necessary to do both. Thus, with the fact that one of two
contradictories must be true, we must know that it exists (is true);

» Anal, past 11. 13. b 15.

* Cf. De aninia, 1. 1, 404 a 9, where ** breathing " is spoken of as the 6pot of ** life," and
the many passages in the Politics where the wora is used to denole that which gives its
special character to the several forms of government (virtue being the tpot of aristocracy,
wealth of oligarchy, liberty of democracy, 1194 a 10) ; Plato, Republic^ viii. 551 c.

' Anal. post. I. 10, 76 a 31 sqq. ^ ibid. Ii. 7, 91 b 10.

* ibid. 91 b 19. * ibid. 91 b 11 sqq. .

144 INTRODUCTION [CB. n. f 7

of the triangle we must know that it means such and such a tiling ; of
the unit we must know botii what it means and that it exists^" What
is here so much insisted on is the very fact which Mill pointed out
in his discussion of earlier views of Definitions, where he says that
the so-called real definitions or definitions of tkin^^ do not constitute
a different kind of definition from nominal definitions, or definitions
of names ; the former is simply the latter pirns something else, namely
a covert assertion that the thii^ defined exists. ''This covert assertion
is not a definition but a postulate. The definition is a mere identical
proposition which gives information only aboiit the use of language,
and from which no conclusion affecting matters of fact can possibly
be drawn. The accompanying postulate, on the other hand, affirms
a fact which may lead to consequences of evenr degree of importance.
It affirms the actual or possible existence of Things possessing the
combination of attributes set forth in the definition : and this, if true,
may be foundation sufficient on which to build a whole fabric of
scientific truth'." This statement really adds nothing to Aristotie*s
doctrine': it has even the slight disadvantage, due to the use of
the word "postulate" to describe "the covert assertion* in all cases,
of not definitely pointing out that there are cases where existence
has to be proved as distinct from those where it must be assmmd.
It is true that the existence of a definiend may have to be taken
for granted provisionally until the time comes for proving it; but,
so far as r^ards any case where existence must be proved sooner
or later, the provisional assumption would be for Aristotie, not a
postulate, but a hypothesis. In modem times, too. Mill's account of
the true distinction between real and nominal definitions had been
fully anticipated by Saccheri*, the editor of Euclides ab omni naroo
vvtdicatus (1733), famous in the history of non-Euclidean geometry.
In his Logica Demofistrativa (to which he also refers in his Euclid)
Saccheri lays down the clear distinction between what he calls de-
finitiones quid nominis or nominaleSy and definitiones quid rei or reales^
namely that the former are only intended to explain the meaning

1 AfuU.posL I. I, 71 a II sqq. ' Mill's System 0/ Logic, Bk. i. ch. viii.

< It is true that it was in opposition to ** the ideas of most of the ArisMelian hgidans**
(rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his
observation: ** We have already made, and shall often have to repeat, the remark, that the
philosophers who overthrew Realism by no means got rid of the consequences of Realism,
but retained long afterwards, in their own philosophy, numerous propositions which could
only have a rational meaning as part of a Realistic system. It had oeen handed down from
Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry
is deduced from definitions. This, so long as a definition was considered to be a proposition
• nnfokling the nature of the thing,* did well enough. But Hobbes followed and rejected
utterly the notion that a definition declares the nature of the thing, or does anything but
state the meaning of a name ; yet he continued to affirm as broadly as any of his predecessors
that the d^oZ, frincipia, or original premisses of mathematics, and even of all science, are
definitions ; producing the singular paradox that systems of scientific truth, nay, all truths
whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of
mankind concerning the signification of words.** ' Aristotle was guilty of no such paradox ;
on the contrary, he exposed it as plainly as did Mill.

^ This has been fuDy brought out in two papers by G. Vailati, La i§cria ArisMdica ddim
dkAmitione (Rwista di FiUsofia e uienu amm\ 1003)1 and Di tm* opera dtmen tu a /a del
P. Gereiamo Saccheri (** Logica DemonstraUva,*' 1(^7) (in Rivista FUiofica^ I903)-

f

r

CH. ix,§7] THE DEFINITIONS 145

that is to be attached to a given term, whereas the latter, besides
declaring the meaning of a word, affirm at the same time the existence
of the thing defined or, in geometry, the possibility of constructing it
The definitio quid nominis becomes a definitio quid rei " by means of a
postulate^ or when we come to the question whether the thing exists and
it is answered affirmatively ^^ Definitiones quid nominis are in them-
selves quite arbitrary, and neither require nor are capable of proof;
they are merely provisional and are only intended to be turned as
quickly as possible into definitiones quid rei, either (i) by means of
a postulate in which it is asserted or conceded that what is defined
exists or can be constructed, e.g. in the case of straight lines and
circles, to which Euclid's first three postulates refer, or (2) by
means of a demonstration reducing the construction of the figure
defined to the successive carrying-out of a certain number of those
elementary constructions, the possibility of which is postulated. Thus
definitiones quid rei are in general obtained as the result of a series of
demonstrations. Saccheri gives as an instance the construction of a
square in Euclid I. 46. Suppose that it is objected that Euclid had
no right to define a square, as he does at the beginning of the Book,
when it was not certain that such a figure exists in nature; the
objection, he says, could only have force if, before proving and making
the construction, Euclid had assumed the aforesaid figure as given.
That Euclid is not guilty of this error is clear from the fact that
he never presupposes the existence of the square as defined until
after i. 46.

Confusion between the nominal and the real definition as thus de-
scribed, i.e. the use of the former in demonstration before it has been
turned into the latter by the necessary proof that the thing defined
exists, is according to Saccheri one of the most fruitful sources of
illusory demonstration, and the danger is greater in proportion to
the "complexity" of the definition, i.e. the number and variety of
the attributes belonging to the thing defined. For the greater is the
possibility that there may be among the attributes some that are
incompatible, i.e. the simultaneous presence of which in a given figure
can be proved, by means of other postulates etc. forming part of the
I basis of the science, to be impossible.

The same thought is expressed by Leibniz also. " If," he says,
" we give any definition, and it is not clear from it that the idea, which
we ascribe to the thing, is possible, we cannot rely upon the demon-
strations which we have derived from that definition, because, if that
idea by chance involves a contradiction, it is possible that even con-
I tradictories may be true of it at one and the same time, and thus our
■ demonstrations will be useless. Whence it is clear that definitions
I are not arbitrary. And this is a secret which is hardly sufficiently
known'." Leibniz' favourite illustration was the " regular polyhedron
with ten faces," the impossibility of which is not obvious at first sight.

^ ** Definitio quid nominis nata est evadere definitio auid rei per postulaium vel dum
j Yenitur ad quaestionem an est et respondetur affirmative.
I * O^cuUs eifragmenis inJdits de Leibniz, Paris, Alcan, 1903, p- 431* Quoted by Vailati.

! H. B. 10

146 INTRODUCTION [CB. dl 1 7

It need hardly be added that, speaking generally, Euclid's defini-
tions, and his use of them, agree with the doctrine of Aristotle
that the definitions themselves say nothing as to the existence of die
things defined, but that the existence of each of them most be
pro>^ or (in the case of the ** prindple^'*) assumed. In geometry,
says Aristotle, the existence of points and lines only must be as*
sumed, the existence of the rest being proved. Accordingly Euclid's
first three postulates declare the possmility of constructing straight
lines and circles (the only "lines except straight lines UMd in the
Elements). Other thin^ are defined and afterwards constructed and
prov«l to exist : e.g. in cook L, Def. 20, it is explained what is meant
by an equilateral triangle ; then (L i ) it is proposed to construct it,
and, when constructed, it is proved to agree with the definitioa
When a square is defined (l. De£ 22), the question whether such a
thing really exists is left open until, in 1. 46, it is proposed to construct
it and, when constructed, it is proved to satisfy the definition^
Similarly with the right angle (L Def. 10, and L ii) and parallels
(L De£ 23, and I. 27 — 29). The greatest care is taken to exclude
mere presumption and imagination. The transition from the sub-
jective definition of names to the objective definition of things is
made, in geometry, by means of coHStmctions (the first principles of
lyhich are postulated), as in other sciences it is made by means ctf .
experience*. }

Aristotle's requirements in a definition.

We now come to the positive characteristics by which, according
to Aristotle, scientific definitions must be marked.

Firsts the different attributes in a definition, when taken separately,
cover more than the notion defined, but the combination of them i
does not Aristotle illustrates this by the " triad," into which enter
the several notions of number, odd and prime, and the last " in both
its two senses (a) of not being measured by any (other) number (ck
M^ fierpelaOai dpi0/i£) and (A) of not being obtainable by adding
numbers together" (o>9 fi)f avyxeurOai i( dpiSfA&v), a unit not being a
number. Of these attributes some are present in all other odd
numbers as well, while the last [primeness in the second sense] \
belongs also to the dyad, but in nothing but the triad are they aU
present •." The fact can be equally well illustrated from geometry.
Thus, e.g. into the definition of a square (Eucl. L, Def. 22) there enter j
the several notions of figure, four-sided, equilateral, and right-angled, |
each of which covers more than the notion into which aU enter ks
attributes!

Secondly, a definition must be expressed in terms of things which
are prior to, and better known than, the things defined*. This is

^ Trenddenbarg, EUmmta Le^ices Aristotdeai^ | 5a

* Trendelenbufg, Erldutenmgm tu den EUfnentm der arisUtelisckgn Lagik^ 3 ed. p. io7.
On constniction as proof of existence in ancient geometry cf. H. G. ZcaXlktskt Die getmeiriselU
Construction als ** ExUtenubem^s ** $n der amtUen GeonutrU (in Matkanatisckg Annalen,
47. Band).

* Anai.past, u. 13, 96 a 33— b i.
^ Trendelenbnig, ErUhtieruneent p. 108. * Tsfia vi. 4, 141 a 96 iqq.

CH. IX.J7] THE DEFINITIONS 147

clear, since the object of a definition is to give us knowledge of the
thing defined, and it is by means of things prior and better known
that we acquire fresh knowledge, as in the course of demonstrations.
But the terms " prior " and " better known " are, as usual susceptible
of two meanings; they may mean (i) absolutely or logically prior and
better known, or (2) better known relatively to us. In the absolute
sense, or from the standpoint of reason, a point is better known than
a line, a line than a plane, and a plane than a solid, as also a unit is
better known than number (for the unit is prior to, and the first
principle of, any number). Similarly, in the absolute sense, a letter is
prior to a syllable. But the case is sometimes different relatively to
us ; for example, a solid is more easily realised by the senses than a
plane, a plane than a line, and a line than a point. Hence, while it is
more scientific to begin with the absolutely prior, it may, perhaps, be
permissible, in case the learner is not capable of following the scientific
order, to explain things by means of what is more intelligible to him,
'^ Among the definitions framed on this principle are £ose of the
point, the line and the plane; all these explain what is prior by
means of what is posterior, for the point is described as the extremity
of a line, the line of a plane, the plane of a solid." But, if it is asserted
that such definitions by means of things which are more intelligible
relatively only to a particular individual are really definitions, it will
follow that there may be many definitions of the same thing, one for
each individual for whom a thing is being defined, and even different
definitions for one and the same individual at different times, since at
first sensible objects are more intelligible, while to a better trained
mind they become less so. It follows therefore that a thing should
be defined by means of the absolutely prior and not the relatively
prior, in order that there may be one sole and immutable definition.
This is further enforced by reference to the requirement that a good
definition must state the genus and the differentiae, for these are
among the things which are, in the absolute sense, better known than,
and prior to, the species (j&v cEttXcS^ yvtoptiumipcuv koX irporipcav rov
tXSav^ iarlp). For to destroy the genus and the differentia is to
destroy the species, so that the former are ^ior to the species ; they
are also better known, for, when the species is known, the genus and
the differentia must necessarily be known also, e.g. he who knows
** man " must also know " animal " and '' land-animal," but it does not
follow, when the genus and differentia are known, that the species is
known too, and hence the species is less known than they are^ It
may be frankly admitted that the scientific definition will require
superior mental powers for its apprehension ; and the extent of its
use must be a matter of discretion. So far Aristotle ; and we have
here the best possible explanation why Euclid supplemented his
definition of a point by the statement in I. Def 3 that the extremities of
a line are points and his definition of a surface by I. Def. 6 to the effect
that the extremities of a surface are lines. The supplementary expla-

* Topics VI. 4, 141 b «5— 34.

148 INTRODUCTION [cH. iz. 1 7

nations do in fact enable us to arrive at a better understanding of the
formal definitions of a point and a line respectively, as is well ex*
plained by Simson in his note on Def. I. Simson says» namelv. that
we must consider a solid, that is, a magnitude wUch has length,
breadth and thickness, in order to understand aright the definitions of
a point, a line and a surfSsice. Consider, for instance, the boundary
common to two solids which are contiguous or the boundary whidi
divides one solid into two contiguous parts; this boundary is a surface.
We can prove that it has no thickness by taking away either solid,
when it remains the boundary of the other; for, if it had thickness, the
thickness must either be a part of one solid or of the other, in which
case to take away one or otiier solid would take away the thickness
and therefore the boundary itself: which is impossible. Thereft)re
the boundary or the surface has no thickness. In exactiy the same
way, r^arding a line as the boundary of two contiguous surfaces^ we
prove that the line has no breadth ; and, lastiy, re^urding a point as
the common boundary or extremity of two lines, we prove that a
point has no length, breadth or thickness.

Aristotie on unscientific definitions.

AristoUe distinguishes three kinds of definition which are un-
scientific because founded on what b n^/ prior (^^ Ic wfHnipm(% The
first is a definition of a 'thin|^ by means of its opposite, eg. of "* good **
by means of " bad " ; this is wrong because opposites are naUirally i
evolved together, and the knowledge of opposites is not uiicommonly I
regarded as one and the same, so that one of the two opposites '■
cannot be better known than tiie other. It is true that, m some
cases of opposites, it would appear that no other sort of definition is
possible: e.g. it would seem impossible to define double apart from the
half and, generally, this would be the case with things which in their
very nature (jcaO' aura) are relative terms {irpi^ n X^ctoa), since one
cannot be known without the other, so that in the notion of either the
other must be comprised as welP. The second kind of definition
which is based on what is not prior is that in which there is a
complete circle through the unconscious use in the definition itself of
the notion to be defined though not of the name*. Trendelenburg
illustrates this by two current definitions, (i) that of magnitude as
that which can be increased or diminished, which is bad because the
positive and negative comparatives "more" and "less" presuppose .
the notion of the positive " great," (2) the famous Euclidean definition ^
of a straight line as that which "lies evenly with the points on itself"
(^f laov roU t4^ iavrvf^ ayfieioi^ K€iT(u\ where " lies evenly " can only ^
be understood with the aid of the very notion of a straight line which is
to be defined^ The tkird kind of vicious definition from that which
is not prior is the definition of one of two coordinate species by means
of its coordinate {dvriSinpfffiivov), e.g. a definition of " odd " as that
which exceeds the even by a unit (the second alternative in Eud. vii.
Def. 7) ; for "odd " and "even " are coordinates, being differentiae of

^ Topia VI. 4, 143 A 13 — 31. * ihid, 143 a 34— b 6.

'* Trendelenbuig, BrULwUntngtH^ P* I'S*

_^ )

1

I

i CH. IX. § 7] THE DEFINITIONS 149

number^ This third kind is similar to the first. Thus, says Tren-
■f delenburg, it would be wrong to define a square as *'a rectangle
\ with equal sides."

Aristotle's third requirement.
' A third general observation of Aristotle which is specially relevant
' to geometrical definitions is that "to know what a thing is (rt ifrriv) is
the same as knowing why it is (S^ rl iffrip)*.'* " What is an eclipse ?
A deprivation of light from the moon through the interposition of the
earth. Why does an eclipse take place? Or why is the moon
eclipsed ? Because the light fails through the earth obstructing it
What is harmony ? A ratio of numbers in high or low pitch. Why
does the high-pitched harmonise with the low-pitched? Because
the high and the low have a numerical ratio to one another*." *' We
seek die cause (to hUtn) when we are already in possession of the
fact {to Sri). Sometimes they both become evident at the same time,
but at all events the cause cannot possibly be known [as a cause]
before the fact is known^'* '' It is impossible to know what a thing is
if we do not know that it is*/* Trendelenburg paraphrases : " The
definition of the notion does not fulfil its purpose until it is made
genetic. It is the producing cause which first reveals the essence of
the thing. ••. The nominal definitions of geometry have only a
provisional significance and are superseded as soon as they are made
genetic by means of construction." Kg. the genetic definition of a
parallelogram is evolved from Eucl. I. 31 (giving the construction for
parallels) and I. 33 about the lines joining corresponding ends of two
straight lines parallel and equal in length. Where existence is proved
by construction, the cause and the fact appear together^

Again, *'it is not enough that the defining statement should set
forth the fact, as most definitions do; it should also contain and
present the cause ; whereas in practice what is stated in the definition
is usually no more than a conclusion (avfiiripaa^). For example,
{I what is quadrature ? The construction of an equilateral right-angled
•I figure equal to an oblong. But such a definition expresses merely the
f conclusion. Whereas, if you say that quadrature is the discovery of a
i mean proportional, then you state the reason ^" This is better under-
i stood if wc compare the statement elsewhere that "the cause is the
'middle term, and this is what is sought in all cases*," and the illustra-
' tion of this by the case of the proposition that the angle in a semi-
circle is a right angle. Here the middle term which it is sought to
establish by means of the figure is that the angle in the semi-circle is
equal to the half of two right angles. We have then the syllogism :
Whatever is half of two right angles is a right angle ; the angle in a
i semi-circle is the half of two right angles ; therefore {conclusion) the
-angle in a semi-circle is a right angle*. As with the demonstration, so

* Topia VI. 4, 143 b 7—10. — • Anal, past, 11. 3, 90 a 31.

* Anal. post. 11. 3, 90 a 15—31. — * ibid, ii. 8, 93 a 17.

* ihid, 9^ a 3a * Trendelenbarg, Erldutemngm, p. 116.
' Dt amma 11. 3, 413 a 13 — 30. * Anal* post. \\. 3, 90 a 6,

* Hid. II. II, 94 a 33.

1

ISO INTRODUCTION [cB.ix.|7

it should be with the definition. A definition idiich is to show the
genesis of the thing defined should contain the middle term or cause ;
otherwise it is a mere statement of a conclusion. Consider, for
instance, the definition of "quadrature" as ''making a square equal in
area to a rectangle with unequal sides." This gnres no hint as to
whether a solution of the problem is possible or how it is solved : bnt,
if you add that to find the mean proportional between two given
straight lines gives another straight une such that the square on it is |
equal to the rectangle contained by the first two straight linesi yoa ■
supply the necessary middle term or cause^. !

Technical terms not defined by Euclid. |

It will be observed that what is here defined, ''quadrature* or
" squaring " (rerpaywviafAosi), is not a geometrical figure, oran attribute
of such a figure or a part of a figure, but a tedmical term used to ,
describe a certain problem. Euclid does not define such things ; but
the fact that Aristotle alludes to this particular definition aswdl as to
definitions of deflection {tc€tc\M0ai) and of verging' {vwlkuf) seems to
show that earlier text-books included among ddinitions explanations
of a number of technical terms, and that Euclid deliberat^ omitted
these explanations from his Elements as surplusage. LaCer the
tendency was again in the opposite direction, as we see from the mudi
expanded Definitions of Heron, which, for example, actually include
a definition of a deflected line (xeKkaafUpff ypa^Lia^nr* Eudid uses the
passive of kX&p occasionally*, but evidently considered it unneoesraiy
to explain such terms, which had come to bear a recognised meaning.
The mention too by Aristotle of a definition of verging (yci/tiy)
suggests that the problems indicated fa^ this term were not excluded
from elementary text-books before Euclid. The type of problem
(vevcisi) was that of placing a straight line across two lines, e.g. two
straight lines, or a straight line and a circle, so that it shall verge to a
given point (i.e. pass through it if produced) and at the same time the
intercept on it made by the two given lines shall be of given length.

Other posages in Aristotle may be quoted to the like effect : e.g. Anai, pott. i. t,
71 b 9 '* We consider that we know a particniar thing in the absolate sense, as distinct
nrom the sophistical and incidental sense, when we consider that we know the cause on
aooonnt of which the thing is, in the sense of knowing that it is the canse of that thing and
that it cannot be otherwise,*' ibid. i. 13, ^pa 1 '* For here to know ihtfact is the fnnctum of!
those who are concerned with sensible thm^ to know the cauu is the ranction of the matbe- ,
matician ; it is he who possesses the proofi of the causes, and often he does not know the
fiict." In view of such passages it is difficult to see how Proclus came to write (p. sos, 11)
that Aristotle was the originator ('A^c^rorAow Korif^pmn) of the idea of Amphmomus ana
others that geometry does not investigate the cause and the wfy {rh Ml W)- To this Geminus
replied that the invertigation of the cause does, on the contrary, appear in geometry. " For
how can it be maintained that it is not the business of the geometer to inauire for wluit reaaon,
on the one hand, an infinite number of equilateral polygons are inscribea in a circle, but, on
the other hand, it is not possible to inscribe in a sphere an infinite numbo' of polyhedral
figures, ejquilateral, equiangular, and made up of similar plane figures? Whose business is it
to ask this question and find the answer to it if it is not that of the geometer? Now when
geometers reason per imfossibiU thejr are content to discover the property, but when they ,
argue by direct proof, it such pool be only partial (M Mpovf)i thb does not suffice for
showing the canse ; if however it is general and applies to all like cases, the why (vt^ M W)
is at once and concurrently made evident."

* Heron, ed. Hultsch, Def. 14, p. 11. * e.g. in III. 90 and in IkOa 89.

IcH. IX.J7] THE DEFINITIONS 151

In general, the use of conies is required for the theoretical solution of
I these problems, or a mechanical contrivance for their practical
j| solution \ Zeuthen, following Oppermann, gives reasons for supposing,
■ not only that mechanical constructions were practically used by the
f older Greek geometers for solving these problems, but that they were
theoretically recognised as a permissible means of solution when the
solution could not be effected by means of the straight line and circle,
and that it was only in later times that it was considered necessary to
use conies in every case where that was possible*. Heiberg* suggests
that the allusion of Aristotle to yevo-ew perhaps confirms this sup-
position, as Aristotle nowhere shows the slightest acquaintance with
conies. I doubt whether this is a safe inference, since the problems
of this type included in the elementary text-books might easily have
been limited to those which could be solved by " plane " methods (i.e.
by means of the straight line and circle). We know, e.g., from Pappus
that Apollonius wrote two Books on plane vevaei^*. But one thing
is certain, namely that Euclid deliberately excluded this class of
Iproblem, doubtless as not being essential in a book of Elements.
Definitions not afterwards used.

Lastly, Euclid has definitions of some terms which he never after-
wards uses, e.g. oblong (irepo/AffKe^), rhombus, rhomboid, trapezium.
The "oblong" occurs in Aristotle; and it is certain that all these
definitions are survivals from earlier books of Elements.

^ Cf. the chapter on vw&vtit in ne IVarks of Archimedes^ pp. c — cxxii.

* Zeuthen, Die Lehre twn den Kegelschniiten im Altertum^ ch. ii, p. 161.
' Heiberg, Mathemaiiseka tu ArisioieUs^ p. 16.

* Pi^pns VII. pp. 670—3.

1

,>■■/

BOOK I.

DEFINITIONS.

•^ I. A point is that which has no part.
<- 2. A line is breadthless length.

3. The extremities of a line are points.

— 4. A straight line is a line which lies evenly with the
points on itself.

— 5* A surface is that which has length and breadth only.

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with
the straight lines on itself.

8. A plane angle is the inclination to one another of
two lines in a plane which meet one another and do not lie in
a straight line.

9. And when the lines containing the angle are straight,
the angle is called rectilineal.

10. When a straight line set up on a straight line makes
the adjacent angles equal to one another, each of the equal
anfi^les is right, and the straight line standing on the other is
called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right
angle.

12. J^ii^iacute angle is an angle less than a right angle.
— 13. A boundary is that which is an extremity of any-
thing.

■^ 14. A figure is* mat which js contained by any boundary
or boundaries.

15. A circle is a plane figure contained by one line such
that all the straight lines falling upon it from one point among
those lying within the figure are equal to one another ;

154 BOOK I [h VIEW. i6— POST. 4

16. And the point is (Ailed the centre of the circle.

1 7. A diameter of the circle is any straight line drawn
through the centre and terminated in both directions by the
circumference of the circle, and such a straight line also
bisects the circle.

1 8. A semicircle is the figure contained by the diameter
and the circumference cut off by it . And the centre of the
semicircle is the same as that of the circle.

19. Rectilineal figures are those which are contained
by straight lines, trilateral figures being those contained by
three, quadrilateral those contained j^ four, and multi-
lateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that
which has its three sides equal, an isosceles triangle that
which has two of its sides alone equal, and a scalene
triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled tri-
angle is that which has a right angle, an obtuse-angled
triangle that which has an obtuse angle, and an acute-
angled triangle that which has its three angles acute.

22. Of quadrilateral figures, a square is that which is
both equilateral and right-angled ; an oblong that which is
right-angled but not equilateral ; a rhombus that which is
equilateral but not right-angled ; and a rhomboid that which
has its opposite sides and angles equal to one another but is
neither equilateral nor right-angled. And let quadrilaterals
other than these be called trapezia.

23. Parallel straight lines are straight lines which,
being in the same plane and being produced indefinitely in
both directions, do not meet one another in either direction.

POSTULATES.

Let the following be postulated :

1. To draw a straight line from any point to any point

2. To produce a finite straight line continuously in a
straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

-L«. . >

^

I- POST. 5— c N. s] DEFINITIONS ETC 155

5* . That, if a straight line falling on two straight lines
make the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right
angles.

COMMON NOTIONS.

1. Things which are equal to the same thing are also
equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders
are equal.

[7] 4- Things which coincide with one another are equal to

one another.

[8] 5. The whole is greater than the part

Definitio^i I.

iilfUUv loTiF, oS fUpot aiOiv,
'^'^ A point is that wAick has no part.

An exactly parallel use of lUfioi {i<rrC) in the sin^lar is found in Aristotle,
Metaph. 1035 b 32 fiipoi fuv cZv Icrrl fcou rov cZSou^ literally *'There is a
/jTf/ even of the form "; Bonitz translates as if the plural were used, *'Theile
giebt es," and the meaning is simply "even the form is divisiMe (into parts)."
Accordingly it would be quite justifiable to translate in this case "A point is
that which is indivisible intoparts.^

Martianus Qipella (5th c. a.d.) alone or almost alone translated differently,
**Punctum est cuius pars nihil t&\^^ *'a point is that a part of which is nothing,^
Notwithstanding that Max Simon {Eudid und die sechs planimitrischen Bilcher^
190 1 ) has adopted this translation (on grounds which I shall presently mention),
I cannot think that it p^ives any sense. If a part of a point is nothings Euclid
Might as well have said that a point is itself ''nothing,'' which of course he
does not do.

Pre-Euclidean definitions.

It would appear that this was not the definition given in earlier text-
books; for Aristotle {Topics vi. 4, 141 b 20), in speaking of *7A^ definitions''
of point, line, atod surface, says that they all define the prior by means of the
posterior, a point as an extremity of a line, a line of a surface, and a surface
of a solid.

The first definition of a point of which we hear is that given by the
Pythagoreans (cf. Proclus^ o. 95, ai), who defined it as a "monad having
position" or ''with position added" (iiova^ wfHxrXaPovau O&riv). It is firequently
wed by Aristotle, either in this exact form (cf. De anima L-4^ 409 a 6) or its
equivalent: e.g. in Metaph, 7016 b 24 he says that that which is indivisible
•very way in respect of ma^ itude and qu& magnitude but has not position is
^wumady while Uiat which is similarly indivisible and has position is 9. paint.

Plato ippMR ID IVPO obfected to this definition. Aristotle says (Meti^h.

156 BOOK I [l

993 a 3o) that he objected "to tfab genua [that of pomti]a$ being a geometrical
fiction {yu^jutrpudw Uy/ia)^ and caUed a point the beginning m a line (4«^
ypa/ifi'^), while again he fre(}uently npoke of 'indiTiaUe lines."* To wbidi
Aristotle replies that even ''indivisil^ lines" most have eitreoiities, so that
the same a^ment which proves the existence of Ufus can be used to prove
ataX points exist It would appear therefore that, when Aristotle objects to
the definition of a point as die extremity of a Une (v^pot ypo^i^) as un-
scientific {Topics VI. 4, 141 b 21X he is aiming at Plato. Heiberg conjectures
{MathemaHsches mu Aristotdes^ p. 8) diat it was due to Plato's influoice that
the word for "point" generally used by Aristode (<myfu|) was r^laoed by
tnuMwv (the r^iular term used by Eudid, Archimedes and later wnters), tfie
latter term {^nota^ a conventional mark) probably hmg conddeied more
suitable than frrvfi»alj (fLpunchtre) whidi mij^ appear to ^lim greater naUiy
for a point

Anstode's conception of a point as diat which is indivisible and has
position is fiirther illustrated by such observations as diat a point is not a
body (De caeh 11. 13, 296 a 17) and has no wri^ {ibiJL in. i, S99 a 30);
again, we can make no distinction between a point and the/Auy (rmf) where
it is {Physics iv. i, 209 a 1 1). He finds the usual difficulty in aooountiii^ for
the transition fix>m the indivisible, or infinitely small^ to the finite or divisible
magnitude. A point being indivisiUe^ no accumulation of points, however fitf
it mav be carried, can give us anything divisiUe, whereas of course a line is a
divisible ma^tude. Hence he holds that points cannot make up anythiiig
continuous like a line, point cannot be continuous with point (00 yap «Eot&v
ixi/uyov arifUiov arifuicv ^ oriyfuf any§KlJ9t -Di gtm» d corr. I. a, 317 a lo), and
a line is not made up of points (ov myMtrw cic frttfium^ Pkysia nr. 8, ai^
b 19). A point, he says, is like the mm in time : nam is indivisiUe and is
not 9k part of time, it is only the b^;inning or end, or a division, of time, and
simikurly a point may be an extremity, b^inning or division of a line, but is
not part of it or of magnitude (cf. De cado in. i, 300 a 14, Pf^sia iv. zi,
220 a I — 21, VI. X, 231 b 6 sqq.). It is only by motion that a point can
generate a line {De anima i. 4, 409 a 4) and thus be the origin of magnitude.

Other ancient definitions.

According to an-NauIzi (ed Curtze, p. 3) one "Herundes" (not so fiu:
identified) defined a point as "the indivisible beginning of all magnitudes,"
• and Posidonius as ''an extremity which has no dimension, or an extremity of
aline."

Criticisms by commentators.

Euclid's definition itself is of course practically the same as that which
Aristotle's fi'equent aUusions show to have been then current, except that it
omits to say that the point must have position. Is it then sufficient, seeing
that there are other things which are without parts or indivisible, e.g. the now
in time, and the unit in number? Proclus answers (p. 93, 18) that the point
is the only thing in the subject-matter of geometry that is indivisible. Relatively
therefore to the particular science the definition is sufficient Secondly^ the
definition has been over and over again criticised because it is purely n^;ative.
Proclus* answer to this is (p. 94, 10) that nq;ative descriptions are appropriate
to first principles, and he quotes Pannenide s as having described his first and
last cause by means of n^;ations merely. Arif totie too admits that it may
sometimes be necessary for one framing a definition to use negations, e.g. in
defining privative terms such as "blind"; and he seems to accept as proper

■I

L DEF. i] NOTE ON DEFINITION i 157

the negative element in the definition of a point, since he says {De anima iii. 6,
430 b 20) that ''the point and every division [e.g. in a length or in a period
of time], and that which is indivisible in this sense, is exhibited as privation

Simplicius (quoted by an-NairIzi) says that ''a point is the beginning of
magnitudes and that from which they grow ; it is also the only thing wluch,
having position, is not divisible.'' He, like Aristotle, adds that it is by its
motion that a point can generate a magnitude : the particular magnitude can
only be "of one dimension," viz. a line, since the point does not "spread
itself" (dimittat). Simplicius further observes that Euclid defined a point
n^atively because it was arrived at by detaching surface from body, line from
surface, and finally point from line. "Since then body has three dimensions

I It follows that a point [arrived at after successively eliminating all three
dimensions] has fwne of the dimensions^ and has no part." This of course
reappears m modern treatises (cf. Rausenbeiger, Eiementar-geometrie des
Punktes^ der Geraden und der Ebtne^ 1S87, p. 7).

An-NairizI adds an interesting observation. " If any one seeks to know
the essence of a point, a thing more simple than a line, let him, in the sensible
world, think of the centre of the universe and the poUs,^^ But there is
nothing new under the sun : the same idea is mentioned, in an Aristotelian
treatise, in controverting those who imagine that the poles have some influence
in the motion of the sphere, "when the poles have no magnitude but are
extremities and points" (De motu animalium 3, 699 a 21).

Modem views.

In the new geometry represented by the excellent treatises which start
from new systems of postulates or axioms, the result of the profound study of
the fundamental principles of geometry during recent years (I need only
mention the names of Pasch, Veronese, Enriques and Hilbert), points come
before lines^ but the vain effort to define them a priori is not made ; instead
of this, the nearest material things in nature are mentioned as illustrations,
with the remark that it is from them that we can get the abstract idea. Cf.
the full statement as regards the notion of a point in Weber and Wellstein,
Encyclopddit der elementaren Mathematik^ 11., 1905, p. 9. "This notion is
evolved from the notion of the real or supposed material point by the process
of limits, Le. by an act of the mind which sets a term to a series of presen-
tations in itself unlimited. Suppose a grain of sand or a mote in a sunbeam,
which continually becomes smaller and smaller. In this way vanishes more
and more the possibility of determining still smaller atoms in the grain of
sand, and there is evolved, so we say, with growing certainty, the presentation
of the point as a definite position in space which is one and is incapable of
further division. But this view is untenable ; we have, it is true, some idea
how the grain of sand gets smaller and smaller, but only so long as it remains
just visible; after that we are completely in the dark, and we cannot see or
imagine the further diminutiop. That this procedure comes to an end is
unthinkable ; that nevertheless diere exists a term beyond which it cannot go,
we must believe or postulate without ever reaching it . . . It is a pure
act of wili^ not of the understanding." Max Simon observes similarly (Euclid^
p. 25) " The notion * point ' belongs to the limit-notions (Grenzbegriffe), the
necessary conclusions of continued, and in themselves unlimited, series of
presentations." He adds, "The point is the limit of localisation; if this is
more and more energetically continued, it leads to the limit-notion 'point,'

158^ BOOK I ' [i. DiTP. I, s

better 'position,' which at the tame time uivohFa a chai^ Content

of space vanishes, relative pasiUm remains. 'Pdnt' theny aocoiding to our
interpretation of Euclid, is the eztremest limit of diat iriiidi we can £il think
of (not observe) as a i;^^/^ presentation, and if we go further than that^ not
only does extension cease but even relative /Auy^ and in this lenie tfie 'put'
is nothing.*^ I confess I think that even the meanii^ which Simon intend! to
convey is better expressed by ''it has 110 part" than uf "the part is nothing,*
since to take a ''part" of a thing in Eudid's tense ii the result of a simple
division, corresponding to an arithmetical fraction, would not be to change
the notion from that of the thing divided to an entirely diflTerent <

f

Definition 2.

TpofLfi^ Sk firJKOi airXarcc

A line is breadihUss length.

This definition may safely be attributed to the Platonic School, if not to
Plato himself. Aristotle {Topies vl 6, 143 b 11) speaks of it as open to
objection because it " divides the goius by n^^ation," loigth being neonsaiily
either breadthless or possessed ofbfeadUi; it woukl seem however that tfie
objection was only taken in order to score a point against the Phtonists, ainoe
he says {ibid. 143 b 29) that the aigument is "of service mtfy against dioae
who assert that the genus [sc. length! is one numerically, diat is^ thoae who
assume ideas^^* e.g. &e idea of lengta (oAri ^i^Kot) which they r^^ard as a
genus : for if the genus, being one and self-existent, could be divided into
two species, one of which asserts what the other denies, it would be sdf-
contradictory (Waitz).

-Proclus (pp. 96, 21 — 97, 3) observes that, whereas the definition of a point
is merely ne^tive, the line introduces the first " dimension," and so its
definition is to this extent positive^ while it has also a n^;ative element which
denies to it the other "dimensions" (Suurrao-ctf). The negation of both
breadth and depth is involved in the single expression "breadthless" (onrXarcg),
since everything that is without breadth is also destitute of depth, though the
converse is of course not true.

Alternative definitions.

The alternative definition alluded to by Produs, ficycfct c^* tr tMurrftror
" magnitude in one dimension " or, better perhaps, " magnitude extended one
way " (since fiuurrao-ic as used with reference to line, sur&ce and solid scarcdy
corresponds to our use of "dimension" when we speak of "one,'' "two," or
"three dimensions"), is attributed by an-NairIzi to "Heromides/* who must
presumably be the same as "Herundes," to whom he attributes a certain
definition of a point It appears however in substance in Aristotie, though
AristoUe does not use the adjective Suurrardp, nor does he apparenUy use
Suunroo-if except of body as having three " dimensions " or " having dimension
(or extension^ oi/ways (vdb^)," the "dimensions" being in his view (i) up
and down, (2) before and behind, and (3) right and left, and "up" being the
principle or beginning of lengthy *< ri^ht " of breadth^ and " before " of d^th
(De caelo ii. 2, 284 b 24). A line is, according to Aristotle, a magnitude
^divisible in one way only" (fun^xv ^Mupcro^X ^ contrast to a ma^tude
divisible in two ways (&X9 ficoi^ror), or a surface, and a magnitude divisible
"in all or in three wa^s" {^riirqi ttal rpi)^ iiaip€r6y\ or a body {Meiaph.
1016 b 25 — 27); or It is a magnitude **cmtintkms one way (or in one
directionX" as compared with magnitudes continuous two ways or thne ways.

^

I. DEF. a] NOTES ON DEFINITIONS i, a 159

which curiously enough he describes as ** breadth " and "depth" respectively
(/Acyctfos 8^ rh fuv i^t* tv ovKcxcf fi'^KOSf rh 8* ^l Suo irX<£ro9, ro 8* ^i rpia piBo^,
Metaph, loao a 11), though he immediately adds that "length " means a line,
" breadth " a surface, and " depth " a body.

Proclus gives another alternative definition as ^^flux of a point '^ (fiwrn
fnujuiiov), i.e. the path of a point when moved. This idea is also alluded to in
Anstotle (De anima i. 4, 409 a 4 above quoted) : " they say that a line by its
motion produces a surface, and a point by its motion a line." "This
definition," says Proclus (p. 97, 8 — 13), "is a perfect one as showing the
essence of the line : he who called it the flux of a point seems to define it
from its genetic cause, and it is not every line that he sets before us, but only
the immaterial line ; for it is this that is produced by the point, which, though
itself indivisible, is the cause of the existence of things divisible."

Proclus (p. 100, 5 — 19) adds the usefiil remark, which, he says, was
current in the school of ApoUonius, that we have the notion of a line when we
ask for the length of a road or a wall measured merely as length ; for in that
case we mean something irrespective of breadth, viz. distance in one
" dimension." Further we can obtain sensible perception of a line if we look
at the division between the light and the dark when a shadow is thrown on
the earth or the moon ; for clearly the division is without breadth, but has
length.

Species of ** lines."

After defining the "line" Euclid only mentions one species of line, the
straight line, although of course another species appears in the definition of a
circle later. He doubtless omitted all classification of lines as unnecessary for
his purpose, whereas, for example. Heron follows up his definition of a line by
a division of lines into (i) those which are " straight " and (2) those which are
not, and a further division of the latter into {a) "circular circumferences,"
(b) "spiral-shaped" (IXucofiScis) lines and {c) "curved" (fcofiirvAat) lines generally,
and then explains the four terms. Aristotle tells us {Metaph. 986 a 25) that
the Pythagoreans distinguished straight (cv^ and curved (fcofiirvAov), and this
distinction appears in Plato (cf. Republic x. 602 q) and in Aristotle (cf. " to a
line belong tiie attributes straight or curved," AnaL post. i. 4i 73 b 19; "as in
mathematics it is useful to know what is meant by the terms straight and
curved," De anima i. i, 402 b 19). But firom the class of "curved" lines
Plato and Aristotle separate off the vcpi^cpiTs or " circular " as a distinct
species often similarly contrasted with straight Aristotle seems to recognise
broken lines forming an angle as one line : thus "a line, if it be bent (kcko/a-
ficn7), but yet continuous, is called one" {Afetaph, 1 01 6 a 2); "the straight line
is more one than the bent line" (ibid. 1016 a 12^. Cf. Heron, Def. 14, "A
broken line («cc«cXa<rficn7 ypofifiif) so-called is a luie which, when produced,
does not meet itse^.**

When Proclus says that both Plato and Aristotle divided lines into those
which are "straight," "circular" (ircpi^c/n;^) or "a mixture of the two," adding,
as regards Plato, that he included in the last of these classes " those which are
caUed helicoidal among plane (curves) and (curves) formed about solids, and
such species of curved lines as arise from sections of solids" (p. 104, i — 5),
he appears to be not quite exact The reference as regards Plato seems to be
to Parmenides 145 b: "At that rate it would seem that the one must have
shape, either straight or round (arrpoyyuXov) or some combination of the two";
but this scarcely amounts to a formal classification of lines. As regards

i6o

BOOK I

[h i«r. t

Aristotle, Produs seems to have in mind the pasage (Ik mcfr i. 9» t68 b i^)
where it is stated that ''all maiiom in qpace, which we call translatioa (4^p£^ is
(in) a straight line, a circle, or a combmation of the two; for the first two are
the only simple (motions)**

For completeness it is desirable to add the substance of Produs* account
of the classification of lines, for frtiich he quotes Geminus as his authority.

Geminus' first classification of lines.

This bqi;ins (p. 1 1 1, i — o) with a division of lines into tmt^osik (otMctoc)
and incomposite (dirvi^cTot). The only illustration given of the iomforiU
class is the *' broken line which forms an angle** (^ «cicXa<r/&^ rai yn way
voiovo-a) ; the subdivision of the huomposiii class then follows (in the test as
it stands the word " composite " is clearly an error for *' incomposite % The
subdivisions of the incompcMite class are repeated in a later passage (pp- 176,
27 — 177, 23) with some additional details. The following diagram reproduces
the effect of both versions as fiu: as possible (all the illustrations mentioned Iqr
Produs being shown in brackets).

Unet

oonipotitc iooonipotite

(broken line fbnmng an ana^) I

forming a figure

or determinate

(drde, ellipse, dssoid)

not forming a fignie

or
indeCenninate

extending witboot limit
(itraia;ht Une, parabola, nypeibola.

I)

The additional details in the second version, which cannot easily be shown
in the diagram, are as follows :

(i) Of the lines which extend without limit, some do not/arm a figure at
aU (viz. the straight line, the parabola and the hyperbola); but some first
*'come together and form a figure ** (Le. have a loop), ''and, for the rest,
extend witibout limit" (p. 177, 8).

As the only other curve, besides the parabola and the hyperbola, which
has been mentioned as proceeding to infinity is the conchoid (of Nicomedes),
we can hardly avoid the condusion of Tannery^ that the curve which has a
loop and then proceeds to infinity is a variety of the conchoid itself. As is

^ Noiapomr tkistoire dis lignes it surf oca ccmrbes dans ftuUiquHi in BnUetim dts sciem^
mathhn, dastronom. t i6r. viii. (1884), PP* '08— 9.

■j

L DEF. a] NOTE ON DEFINITION a x6i

well known, the ordinary conchoid (which was used both for doubling the
cube and for trisecting the angle) is obtained in this way. Suppose any
number of rays passing through a fixed point (the pole) and intersecting a
fixed straight line ; and suppose that points are taken on the rays, beyond the
fixed straight line, such that the portions of the rays intercept^ between the
fixed straight line and the point are equal to a constant distance (iidtmifia),
the locus of the points is a conchoid which has the fixed straight line for
asymptote. If the "distance " a is measured fi-om the intersection of the ray
with the given straight line, not in the direction awa^ from the pole, but
towards the pole, we obtain three other curves accordmg as a is less than,
equal to, or greater than ^, the distance of the pole from the fixed straight line,
which is an asymptote in each case. The case in which a>d gives a curve
which forms a loop and then proceeds to infinity in the way Proclus describes.
Now we know both from Eutocius (Comm. an Archimedes^ ed. Heiberg, iii.
p. 114) and Proclus (p. 272, 3 — 7) that Nicomedes wrote on conchoidr (in
the plural) and Pappus (iv. p. 244, 18) says that besides the "first" (used as
above stated) there were "the second, the third and the fourth which are
useful for other theorems.''

(2) Proclus next observes (p. 177, 9) that, of the lines which extend
without limit, some are ^^ asymfioiic** (davfwrTWToi), namely "those which
never meet, however they are produced," and some are ^^ symptotic^** namely
"those which will meet sometime"; and, of the "asymptotic" class, some
are in one plane, and others not Lastly, of the "asymptotic" lines in one
plane, some preserve always the same distance firom one another, while others
continually "lessen the distance, like the hyperbola with reference to the
straight line, and the conchoid with reference to the straight line."

^ Geminus' second classification.

I This (from Proclus, pp. iii, 9 — 20 and 112, 16—18) can be shown in a
diagram thus :

Incomposite lines
I dffit^BiTOi ypa/iiuU

f I ' I

' simple, irXfj mixed, fuicHj

1 . I

making a figure indeterminate

(e.g. circle) (straight line)

I ' 1

lines in planes lines on solids

I td h roit CT€ptott

n

line meeting itself extending without limit

(e.g. cissoid)

lines formed by sections lines round solids

aX Karii rdt roftdt al w€pl rd rr€p€d

(e.g. conic sections, spiric curves) (e.g. heiix about a sphere or about a cone)

(cylindrical helix) (all others)

Notes on classes of "lines" and on particular curves.

We will now add the most interesting notes found in Proclus with
reference to the above classifications or the particular curves mentioned.

H. E. II

L

x6a BOOK I [i. DV. t

1. Homoeomeric lines.

By this tenn {ifunofuptU) are meant lines which are alike in all partii so
that in any one such curve any part can be made to coincide with any odier
part Proclus observes that these lines are only three in number, two being
''simple'' and in a plane (the stra^ht line anid the circle^ and the dura
''mixed," (subsisting) "alxMit a sohd,** namdy the cylindrical hdiz. The
latter curve was also called the nxkiias or tufckUom^ and its k^moemmnc
property was proved by ApoUonius in his woric wiyn tov K^xkiom (Prochis,
p. 105, 5). The fact that mere are only three kamoeomiru lines was proved
by Geminus, "who proved, as a preliminary proposition, that, if from a point
{M Tov o^ficiov, but on p. 251, 4 iu^ Mt oiyfMiov) two stnuf^t lines be orawn
to a homoeomeric line making equal angles with it, the straight Hnei are
equal" (pp. 112, 1— 113, 3. cfc P- ^S't 2—19)-

2. Mixed lines.

It might be supposed, says Proclus (p. 105, xxX that the cylindrical hdiz,
being homoeomeric^ like the straight liiMS and the cirde, must like them be
simple. He replies that it is not smiple, but mixei^ because it is generated by
two unlike motions. Two like motions, said Geminus, eg. two motions at the
same speed in the directions of two adjoining sides of a sc^uare, produce a
simple hne, namely a straight line (the diagonal); and again, if a straight line
moves with its extremities upon the two sides of a right an^e respectively, i
this same motion gives a simpk curve (a circle) for tte locus of the midme J
point of the straight line, and a mUxii curve (an dlipae) for the locus of any I
other point on it (p. 106, 3 — 15). i

Geminus also explained that the term ^ mixed," as ai^lied to curves, and
as applied to surfaces, respectively, is used in different senses. As apf^ed to |
curves, "mixing" neither means simple "putting together ** ((nMco-it) nor
" blending " (Kpcuric). Thus the helix (or spiral) is a " mixed " Une, but (i) it j
is not " mixed '* in the sense of "putting together," as it would be if, say, part ]
of it were straight and part circular, and (2) it is not mixed in the sense of
" blending," beoiuse, if it is cut in any way, it does not present the appearance
of any simple lines (of which it might be supposed to be compounded, as it \
were). The " mixing " in the case of lines is rather that in which the con-
stituents are destroyed so far as their own character is concerned, and are
replaced, as it were, by a chemical combination {yrrw iv avrj avpt^apfiiva rk
oKpa KoX inrvK€xiffjJva). On the Other hand " mixed " surfaces are mixed in
the sense of a sort of " blending " (icara rti^a Kpcuriv). For take a cone gene- '
rated by a straight line passing through a fixed point and passing always j
through the circumference of a circle : if you cut this by a plane puallel to ^
that of the circle, you obtain a circular section, and if you cut it by a plane
through the vertex, you obtain a triangle, the " mixed " surface of the cone |
being thus cut into simple lines (pp. 117, 22 — 118, 23).

3. Spiric curves.

These curves, classed with conies as being sections of solids, were dis-
covered by Perseus, according to an epigram of Eratosthenes quoted by
Proclus (p. 112, i), which says that Perseus found "three lines upon (or,
perhaps, m addition to) five sections" (rpcic ypa/ifia« M wivT€ to/muv).
Proclus throws some light upon these in the following passages :

"Of the spiric sections, one is interfaced, resembling the horse-fetter j
{hntov wi&ti) ; another is widened out in the middle and contracts on each

1!

I.DEF. a] NOTE ON DEFINITION d 163

side (of the middle), a third is elongated and is narrower in the middle,
broadening out on each side of it" (p. 112, 4 — 8).

"This is the case with the spiric sutface \ for it is conceived as generated
by the revolution of a circle remaining at right angles [to a plane] and turning
about a point which is not its centre [in other words, generated by the revo-
lution of a circle about a straight line in its plane not passing through the
centre]. Hence the spire takes three forms, for the centre [of rotation] is
either on the circumference, or within it, or without it. And if the centre of
rotation is on the circumference, we have the continuous spire (owfxi;?), if
within, the interlaced (ifjLV9ir\€yfuvyj\ and if without, the open (Sicxi/f). And
the spiric sections are three according to these three differences'' (p. 119,
8-17).

" When the Aippopede, which is one of the spiric curves, forms an angle
with itself, this angle also is contained by mixed lines" (p. 127, i — 3).

"Perseus showed for spirics what was their property ((rvfiirrcDfui) *'
(P- 356, ").

Thus the spiric surface was what we call a tore, or (when open) an anchor-
ring. Heron (Def 98) says it was called alternatively spire (cnrcipa) or ring
(fcpocos); he calls the variety in which "the circle cuts itself," not "interlaced,"
but "crossing-itself" (ffvaXXarrovo-a).

Tannery^ has discussed these passages, as also did Schiaparelli*. It is clear
that Proclus' remark that the difference in the three curves which he mentions
corresponds to the difference between the three surfaces is a slip, due perhaps
to too hurried transcribing from Geminus : all three arise from plane sections
of the open anchor-ring. If r is the radius of the revolving circle, a the
distance of its centre from the axis of rotation, d the distance of the plane
section (supposed to be parallel to the axis) from the axis, the three curves
described in the first extract correspond to the following cases :

(i) d^a-r. In this case the curve is the hippopede^ of which the
lemniscate of Bernoulli is a particular case, namely that in which a = 2r.

The name hippopede was doubtless adopted for this one of Perseus' curves
on the ground of its resemblance to the hippopede of Eudoxus, which seems to
have been the curve of intersection of a sphere with a cylinder touching it
internally.

i2) fl + r > //><!. Here the curve is an ovaL
3) a>d>a-r. The curve is now narrowest in the middle.
Tannery explains the "three lines upon (in addition to) five sections"
thus. He points out that with the open tore there are two other sections
corresponding to

(4) d= a : transition from (2) to (3).

(5) fl-r>^>o, in which case the section consists of two symmetrical
ovals.

He then shows that the sections of the closed or continuous tore^ corre-
sponding to a = r, give curves corresponding to (2), (3) and (4) only. Instead
of (i) and (5) we have only a section consisting of two equal circles touching
one another.

On the other hand, the third spire (the interlaced variety) gives three new
forms, which make a group of three in addition to the first group oifive sections.

> Pour rhistoire da Hgtus et surfaces courbts dams rasttiquitl in BulUtin des sciences
mathim, et astronom. vili. (1884), pp. 35 — 37.

' Die kofmoceniriscken Spkarm des Eudoxus^ des Kallippus und des ArisioteUs (Ahhrnnd-
lungm tstr Geuk, der Maik. I. Heft, 1877, pp. 149 — 151.

II — 2

L-—

164

BOOK I

[l. DEP.

The difficulty which I see in this interpretation is the fiict that, just after
"three lines on five sections " are mentioned, Produs describes three curves
which were evidently the roost important ; but these three belong to three of
the five sections of the open tore^ and are not separate from them.

4. The cissoid.

This curve is assumed to be the same as that by means of which, accordirig
to Eutodus {Comm. on Atxkimiies^ in. p. 79 sqq.^ Diodes in his book w^
irvfMiv {On iuming-gkuses) solved the problem of doubling the cube. It is
the locus of points which he found b^ the following construction. Let AC^
BD be diameters at right angles in a arcle with centre A

Let E^ J^be points on the quacbants BC^ BA respectivdy such that the
arcs BE^ BE axe equal.

Draw EG, Elf perpendicular to CA.
Join AE^ and let F be its intersecticm
with Eir.

The dssoid is the locus of all the
points E corresponding to different posi-
tions of E on the quadrant BC and of E
at an equal distance from B along the arc
BA.

^ is the pointy the curve correspond-
ing to the position C for the point JE*, and
B the point on the curve correqKmding
to the position of E in which it ooinddes
with A

It is easy to see that the curve extends
in the direction AB beyond B, and that
CJ^ drawn perpendicular to CA is an
asjrmptote. It may be regarded also as
having a branch AD symmetrical with
ABf and, beyond D^ approaching EC produced as asjrmptote.

If OA, OD are coordinate axes, the equation of the curve is obviously

where a is the radius of the drcle.

There is a cusp at A, and it agrees with this that Proclus should say
(p. 126, 24) that "dssoidal lines converging to one point like the leaves of
ivy — for this is the origin of their name — form an angle." He makes the
slight correction (p. 128, 5) that it is not two parts of a curve, but am curve,
which in this case makes an angle.

But what is surprising is that Produs seems to have no idea of the curve
passing outside the drcle and having an asymptote, for he several times
speaks of it as a closed curve (forming a figure and including an area): cf.
p. 152, 7, "the plane (area) cut off by the cissoidal line has one bounding
(line), but it has not in it a centre such that all (straight lines drawn to the
curve) fix>m it are equal." It would appear as if Proclus regarded the cissoid
as formed by ^efour symmetrical dssoidal arcs shown in the figure.

Even more pecub'ar is Proclus' view of the

5. *• Single-turn Spiral."

This is really the spiral of Archimedes traced by a point starting fix>m
the fixed extremity of a straight line and moving uniformly along it, while

1

I. DEFF. 2—4] NOTES ON DEFINITIONS 2—4 165

simultaneously the straight line itself moves uniformly in a plane about its fixed
extremity. In Archimedes the spiral has of course any number of turns, the
straight line making the same number of complete revolutions. Yet Proclus,
while giving the same account of the generation of the spiral (p. 180, 8 — 12),
regards the single-turn spiral as actually stopping short at the point reached
after one complete revolution of the straight line : " it is necessary to knaw
that extending without limit is not a property of all lines; for it neither
belongs to the circle nor to the cissoid, nor in general to lines which form
figures ; nor even to those which do not form figures. For even the single-
turn spiral does not extend without MmiX— far it is constructed between two
points — nor does any 0/ the other lines so generated do so'* (p. 187, 19 — 25).
It is curious that Pappus (viii. p. mo sqq.) uses the same term fiovoa-Tpo^
2Ai^ to denote one turn, not of the spiral, but of the cylindrical helix.

Definition 3.

Tpofjififj^ ik wipara irqiiMia,

The extremities of a line are points.

It being unscientific, as Aristotle said, to define a point as the '* extremity
of a line " («pas ypafifi^^)^ thereby explaining the prior by the posterior,
Euclid defined a point differently; then, as it was nec^^Bary to connect a
point with a line, he introduced this escplanation after the definitions of both
had been given. This compromise is no doubt his own idea; the same
thing occurs with reference to a surface and a line as its extremity in Def. 6,
and with reference to a solid and a surface as its extremity in xi. Def. 2.

We miss a statement of the facts, equally requiring to be known, that a

** division " ^Stcupco-ts) of a line, no less than its " begmning ** or *' end," is a

point (this is brought out by Aristotle: cf. Jfetaph, 1060 b 15), and that

the intersection of two lines is also a point If these additional explanations

had been given, Proclus would have been spared the difiSculty which he finds

in the fact that some of the lines used in Euclid (namely infinite straight lines

on the one hand, and circles on the other) have no "extremities." So also

the ellipse, which Proclus calls by the old name Bvptos (" shield "). In the

case of the circle and ellipse we can, he observes (p. 103, 7), take a portion

I bounded by points, and the definition applies to that portion. His rather

I far-fetched distinction between two aspects of a circle or ellipse as a line and

* as a closed figure (thus, while you are describing a circle, you have two extremi-

< ties at any moment, but they disappear when it is finished) is an unnecessarily

elaborate attempt to establish the literal universality of the "definition,"

which is really no more than an explanation that, if a line has extremities,

those extremities are points.

Definition 4.

A straight line is a line which lies evenly with the points on itself

The only definition of a straight line authenticated as pre-Euclidean is
that of Plato, who defined it as " that of which the middle covers the ends "
(relatively, that is, to an eye placed at either end and looking along the
straight line). It appears in the Farmenides 137 s : "straight is whatever has
its middle in fi'ont of (Le. so placed as to obstruct the view of) both its ends "

x66 BOOK I [i.Dir.4

(cMv y€ ov ivri fuo-ovififi^o&roir J^TOCV.Mvpotf^crf). Alistode ^poCtt it in
equivalent terms (72^/W vi. ix, 148 b 2j\ oS ri jUnv iw u rfi 9 $ u row wipmn;
and, as he does not mention the name of its author, but states it in combina-
tion with the definition of a line as the extremity of a surbce, we vm assume
that he used it as being weU known. Produs also quotes the definition as
ipiato's in almost identictd termSi' i% ra gUm roUiKpoii9iw a r pa o$u(p. X091 ai).
This definition is ingenious, but implicitly appeals to the sense of sight and
involves the postulate that the line of sight is straiffht (Cf. the A ri stotel i an
Problems 31, 20, 959 a 39, where the question is why we can better observe
straightness in a row, say, of letters with one eye than with twa) As nqguds
the straightness of ''visual iEys,'*a^cii^ d Euclid's own Q^tks, De£ x, a,
assumed as hypotheses^ in which he first speaks of the ''stnuj^t lines" drawn
from the eye, avoiding the word jfcis, and then sa^ that the figure contained
by the visual rays (o^ces) is a cone with its vertex m the eye«

As Aristotle mentions no definition of a straight line resembling EucUd's,
but gives only Plato's definition and the other explaining it as the ''extremity
of a surface,'' the latter being evidently the current defimtion in contemporary
textbooks, we may safely infer that Euclid's definition was a new departure oif
his own.

Proclus on Euclid's definition.

Coming now to the interpretation of Euclid's definition, cMm yni^
iariVf ipif i( wov roi« i^* javr^ mffuioft kuto^ we find any number of sUghtly
different versions, but none that can be described as quite satisfiictory ; some
authorities, e.g. Savile, have confessed that they could make nothing of it It
is natural to appeal to Proclus first ; and we find that he does in net give an
interpretation which at first sight seems plausible. He says (p. X09, 8 sq.) that
Euchd "shows by means of this that the straight line alone [of all lines]
occupies a distance (icarcxciv Su^on^/ia) equal to that between the points on it
For, as fiar as one of the points is distant from another, so great is the length
(f&^yc^os) of the straight line of which they are the extremities ; and this b the
meaning of lying i( urov to (or with) the points on it " \i£ ttrw being thus, J
apparently, interpreted as "at" (or "over") "an equal distance"]. "But if \
you take two points on the circumference (of a circle) or any other line, the f
distance cut off between 'them along the line is greater than the interval
separating them. And this is the case with every line except the straight line.
Hence the ordinary remark, based on a common notion, that those who
journey in a straight line only travel the necessary distance, while those who
do not go straight travel more than the necessary distance." (Cf. Aristotle, j
De catlo i. 4, ayi a 13, "we always .call 'the distance of anything the straight 1
line " drawn to it) Thus Proclus would interpret somewhat in this way : "a J
straight line is that which represents extension equal with (the distances
separating) the points 'on it" This explanation seems to be an attempt to i
graft on to Euclid's definition the assumption (it is a Aofi/Savd/icvor, not a
definition) of Archimedes {On the sphere and cylinder i. ad init) that "of all <
the lines which have the same extremities the straight line is least" For this ,
purpose li Zcrov has apparently to be tiCken as meaning "at an equal distance,"
and again "lying at an e<^ual distance" as equivalent to "extending over (or *
representing) an equal distance." This is difficult enough in itself, but is
seen to be an impossible interpretation when applied to the similar definition
of a plane by Euclid (Def. 7) as a surfece "which lies evenly with the straight
lines on itself." In that connexion Proclus tries to make the same words If Zirou

t

J

'

I. DEF. 4] NOTE ON DEFINITION 4 167

Mirai mean "extends over an equal area with." He says namely (p. 117, 2)
that ''if two straight lines are set out" on the plane, the plane surface
'' occupies a space equal to that between the straight lines." But two straight
lines do not determine by themselves any space at all ; it would be necessary
to have a closed figure with its boundaries in the plane before we could arrive
at the equivalent of the other assumption of Archimedes that ''of surfaces
which have the same extremities, if those extremities are in a plane, the plane is
the least [in areal." This seems to be an impossible sense for l^ txrov even on.
the assumption tnat it means "at an equal distance" in the present definition.
The necessity therefore of interpreting li Sorov similarly in both definitions
makes it impossible to regard it as referring to distance or length at all. It
should be added that Simplicius gave the same explanations as Proclus
(an-Nairld, p. 5).

' The language and construction of the definition.

Let us now consider the actual wording and grammar of the phrase igfris ii
WW TOic i^ Iavn7f mffjAioi^ Kcirai. As regards Sxe expression ii lo-ov we note
that Plato and Aristotle (whose use of it seems typical^ commonly have it in
the sense of "on a footing of equality": cf. 61 1( law m Plato's Laws 777 d,
919 d; Aristotle, Politics 1259 b 5 l{ lo-ov ctvat Povkerai r^v ^vcriv, "tend to
be on an equality in nature," £tA, Nic. viii. 12, 1 161 a 8 IvravOa a-ovrcc i^
Iffw^ " there all are on a footing of equality." Slightly different are the uses
in Aristotle, Eth. Nic. X. 8, 11 78 a 25 rcSv /icv y^ avayKotW xp«ta ical l{ Ztrov
loTo^ "both need the necessaries of life to the same extent^ let us say"; Topics ix.
15, I74a32^ lirov woiovvra rrfv ipiDTtfiny, "asking the question indifferently"
(i.e. without showing any expectation of one answer being given rather than
another). The natural meaning would therefore appear to be "evenly placed"
(or balanced), "in equal measure," "indifferently" or "without bias" one way
or the other. Next, is the dative roU ^^* iavrq^ (n;/icioi$ constructed with i( urov
or with Kfirai? In the first case the phrase must mean "that which lies evenly
with (or in respect to) the points on it," in the second apparently "that which,
in (or by) the points on it, lies (or is placed) evenly (or uniformly)." Max Simon
takes the first construction to give the sense "die Gerade liegt in gleicher
Weise wie ihre Punkte." If the last words mean " in the same way as (or in
like manner as) its points," I cannot see that they tell us anything, although
Simon attaches to the words the notion o( distance (Abstand) like Proclus.
The second construction he takes as giving "die Gerade liegt fiir (durch) ihre
Punkte gleichmassig," "the straight line lies symmetrically for (or through) its
points"; or, if «c€irai is taken as the passive ofri^/u, "die Gerade ist durch
ihre Punkte gleichmassig gegeben worden," "the straight line is symmetrically
determined by its points." He adds that the idea is here direction, and that
both direction and distance (as between two different given points simply)
would be to Euclid, as later to Bolzano (Betrachtungen Oder einige Gegenstdnde
der Elementargeometriey 1804, quoted by Schotten, Inhalt und Methode des
planimetrischen Unterrichts, 11. p. 16), primary irreducible notions.

While the language is thus seen to be hopelessly obscure, we can safely
say that the sort of idea which Euclid wished to express was that of a line
which presents the same shape at and relatively to all points on it, without
any irr^ular or unsymmetrical feature distinguishing one part or side of it
from another. Any such irr^ularity could, as Saccheri points out (Engel and
Stackel, Die Theorie der ParalleUinien von Euklid bis Gauss, 1895^ p. 109X be
at once made perceptible by keeping the ends fixed and turning the line about

i68 BOOK I [ldbf.

them right round; if any two positions were distinguishable, e.g. one being to
the left or right relatively to another, "it would not lie in a uniform manner
between its points."

A conjecture as to its origin and meaning.

The question arises, what was the origin of Euclid's definition, or, how
was it suggested to him ? It seems to me that the basis of it was really
Plato's definition of a straight line as "that line the middle of which covers
the ends." Euclid was a Platonist, and what more natural than that he
should have adopted Plato's definition in substance, while regarding it as
essential to change the form of words in order to make it independent of any
implied appeal to vision, which, as a physical fiict, could not properly find a
place in a purely geometrical definition? I believe therefore that Eudid's
definition is simply an attempt ^albeit unsuccessful, fix>m the nature of the
case) to express, in terms to which a geometer could not object as not being
part of geometrical subject-matter, the same thing as the Platonic definition.

The truth is that Euclid was attempting the impossible. As Pfleiderer
says (Scholia to Euclid), " It seems as though the notion of a straight Ime^
owing to its simplicity, cannot be explained by any regular definition which
does not introduce words already containing m themsislv^ by implication,
the notion to be defined (such eg. are duection, equali^, uniformity or
evenness of position, unswerving course), and as though it were impossible^ if
a person does not already know what the term siraigki here means, to teadi
it to him unless by putting before him in some way a pictuie or a dtrawing of
it" This is accordingly done in such books as Veronese's Eiemtnii H
geofnetria (Part i., 1904, p. 10): ''A stretched string,* e.g. a plumniet, a ray of
light entering by a small hole into a dark room, are rutiUmal objects. The
image of them gives us the abstract idea of the limited line which is called a
rectilineal segment!^

Other definitions.

We will conclude this note with some other fan^ous definitions of a straight
line. The following are given by Proclus (p. no, 18—23).

I. A fine stretched to the utmost^ itr oKpov rcro/icn; ypa/A/Aif. This appears
in Heron (ist c a.d.) also, with the words "towards the ends" (hri ra a^^ra)
added. (Heron, ed. Hultsch, Def. 5, p. 8).

a. Part of it cannot he in the assumed plane while part is in one higher up
i^ fur€iap€T4pff). This is sl proposition in Euclid (xi. i).

3. jiU its parts Jit on all (other parts) alike, wavra avrfs rot fUfni www
ofiolia^ iffMpfwiti, Heron has this too (Def. 5), but instead of "alike" he
says iravTotm, "in all ways," which is better as indicating that the applied part
may be applied one way or the reiferse way, with the same result

4. That line which^ when its ends remain fixed^ itself remains fixed^ ri rw
w€paTW¥ fLcvoWttiv icttt aMi fiivovau. Heron's addition to this, " when it is, as
it were^ turned round in the same plane*^ (olov h rf avnp lwvw&^ irrp€^ofUyti\
and his next variation, "and about the same ends having always the same
position," show that the definition of a straight line as "that which does
not change its position when it is turned about its extremities (or any two
points in it) as poles" was no original discovery of Leibniz, or Saccheri, or
Krafll, or Gauss, but goes back at least to the banning of the Christian era.
Gauss' form of this definition was: "The line in which lie all points that,
during the revolution of a body (a part of space) about two fixed points,
maintain their position unchanged is called a straight line." SchoUen

7\

1. DEFr. 4, S] NOTES ON DEFINITIONS 4. S 169

(i* P- 315) maintains that the notion of a straight line and its property of
being determined by two points are unconsciously assumed in this definition,
which is therefore a logical ''circle/'

5. Tkat line which with one other of the same species cannot complete a
figure^ 17 /xrra r^s ^fiociSovs fuas o-x/ffAa firi dmrtkowra. This is an obvious
SaT€poy'irp6T€poy, since it assumes the notion of a figure.

Lastly Leibniz' definition should be mentioned: A straight line is one
which divides a plane into two halves identical in all but position. Apart from
the fact that this definition introduces the plane, it does not seem to have any
advantages over the definition last but one referred to.

Legendre uses the Archimedean property of a straight line as the shortest
distance between two points. Van Swinden observes {Elemente der Geometries
1^341 P- 4)1 that to take this as the definition involves axjf#m//7^ the'proposition
that any two sides of a triangle are greater than the third and proving that
straight lines which have two points in common coincide throughout their
length (cf- Legendre, Aliments de Geometric 1. 3; 8).

The above definitions all illustrate the observation of Unger {Die Geometric
des Euklidy 1833) : ^^ Straight is a simple notion, and hence all definitions of
it must fail.... But if the proper idea of a straight line has once been grasped,
it will b^ recognised in all the various definitions usually given of it ; all
the definitions must therefore be regarded as explanations^ and among them
that one is the best from which further inferences can immediately be drawn
as to the essence of the straight line."

Pefinition 5.

A surface is that which has length and breadth only.

The word ^irt^vcui was used by Euclid and later writers to denote surface
in general, while they appropriated the word cViirffSov for plane surface, thus
making htlw&w a species of the gentis iwif^tia, A sob'ts^ use of cn-i^oFcia
by Euclid when a plane is meant (xi. Def. 1 1) is probably due to the fact that
the particular definition came from an earlier textbook. Proclus (p. 116, 17)
remarks that the older philosophers, including Plato and Aristotle, used the
words lwi^v€ia and criircSov indifferently for any kind of surface. Aristotle
does indeed use both words for a surface, with perhaps a tendency to use
ciri^aycia more than criVcSov for a surface not plane. Cf. Categories 6, 5 a i sq.,
where both words are used in one sentence: "You can find a common
boundary at wiiich the parts fit together»,a point in the case of a line, and a line
in the case of a surface (cin^Vcia); for the parts of the surface (cViircSov) do fit
together at some common boundary. Similarly also in the case of a body you
can find a common boundary, a line or a surface (ciri^oi^cia), at which the
parts of the body fit together." . Plato however does not use ciri^Vcca at all in
the sense of surface, but only cv-iircSov for both surface and plane surfau.
There is reason therefore for doubting the correctness of the notice in
Diogenes Laertius, in. 24, that Plato "was the first philosopher to name,
among extremities, ^'t plane surface " (cViircSos cv-c^ovcta).

ffVi^cca of course means literally the feature of a body which is apparent
to the eye (cirt^nfc), namely the surface.

Aristotle tells us {De sensu 3, 439 a 31) that the Pythagoreans called a
surface XP®^ which seems to have meant skin as well as colour, Aristotle
explains the term with reference to colour (xp«i)f^) as a thing inseparable from
the extremity (ir^«) of a body.

X70 BOOK I

Alternative definitions.

The de6nitions of a surfiu» correspond to diose of a line. As in Aristotle
a line is a magnitude ''(extended) one way, or in one 'dimension'** (c^* V^
"continuous one way" (c^' tr av¥€xk\ or *' divisible in one way" OMpaxJ
Scoipcroi'), so a surface is a masnitude extended or continuous iwa ways (m
Svo), or divisible in two ways (oixp). As in Euclid a surftce has "length and
breadth '' only, so in Aristotle " breadth " is characteristic of the surface and is
once used as synonymous with it (Meiafh. X020 a laX and again ''lengths
are made up of long and short, surfaas of broad amd narrow^ and solids (<7km)
of deep and shallow " (Meiaph. 1085 a 10).

Anstotle mentions the common remark that a Une by Us moHon produces a
surface {De anima i. 4, 409 a 4). He alsogives the a posteriori deKripti<Hi of
a surface as the "extremitv of a solid** (Topics vi. 4, 141 b aa), and as "the
section (rofiif) or division (fiuJpmn^) of a bo^" (ifetaph* 1060 b 14).

Proclus remarks (p. 114^ ao) that we get a notion of a surfiure when we
measure areas and mark their boundaries in the sense of length and breadth ;
and we further get a sort of perception of it by looking at shadows, since
these have no depth (for they do not penetrate the earth) but only have leqg^
and breadth. * -

Classification of surfaces.

Heron gives (Def. 75, p. 23, ed. Hultsch) two alternative divisions of
surfaces into two classes, corresponding to Geminus* alternative divisions of
lines, viz. into (i) incomposite and composite and (2) single and mixed.

(i) Incomposite surfaces are ^ those which, when produced, fidl into (or
coalesce with) themselves" (&rac UfioXkiiLwu a^roi tnJf iavni^ wArroiwtr),
i.e. are of continuous curvature, e.g. the sphere.

Composite surfaces are "those which, when produced, . cut one another."
Of composite surfaces, again, some are (a) made up of non-homogeneous
(elements) (c^ &,vo/ioiay€y£v) such as cones, cylinders and hemispheres, others
lb) made up of homogeneous (elements), namely the rectilineal (or polyhedral)
surfaces.

(2^ Under the alternative division, simp/e surfaces are the plane and the
sphencal surfaces, but no others ; the mixed class includes all other sur&ces
whatever and is therefore infinite in variety.

Heron specially mentions as belonging to the mixed class (a) the surfoce
of cones, cylinders and the like, which are a mixture of plane and circular
(fjLucraX ff^ cirtirffSov koI vcpt^cpcm) and (b) spiric surfaces, which are "a mixture
of two circumferences " (by which he must mean a mixture of two drcular
elements, namely the generating circle and its circular motion about an axis in .
the same plane).

Proclus adds the remark that, curiously enough, mixed surfaces may arise*
by the revolution either of simple curves, e.^. in the case of the spire^ or of
mixed curves, e.g. the "right-angled conoid" from a parabola, "another
conoid" from the hyperbola, the "oblong" (c«>cffiyiccv, in Archimedes vapa-
fioiccs) and " flat " (ImrAarv) spheroids from an ellipse according as it revolves .
about the major or minor axis respectively (pp. 119, 6 — 120, 2). The hamoeo-
meric surfaces, namely those any part of which wiU coincide with any other
part, are two only (the plane and the spherical surface), not three as in the case
of lines (p. 120, 7).

J

1. DBFF. 6, 7] NOTES ON DEFINITIONS 5—7 171

Definition 6.

*Eiri^aFffia« Sc wipara ypCLfjifAoL

The extremities of a surface are lines.

It being unscientific, as Aristotle says, to define a line as the extremity of
a surface, Euclid avoids the error of defining the prior by means of the
posterior in this way, and gives a different definition not open to this
objection. Then, by way of compromise, and in order to show the connexion
between a line and a surface, he adds the equivalent of the definition of a line
previously current as an explanation.

As in the corresponding Def. 3 above, he omits to add what is made
dear by Aristotle {Metaph. 1060 b 15) that a "division" (Staipco-is) or
"section" (ro/ii;) of a solid or body is also a surface, or that the common
boundary at which two parts of. a solid fit together {Categories 6, 5 a 2)
may be a surface.

Proclus discusses how the fact stated in Def. 6 can be said to be true of
surfaces like that of the sphere "which is bounded (ircirffpa<rraO, it is true, but
not by lines." His explanation (p. 116, 8 — 14) is that "if we take the surface
(of a sphere), so far as it is extended two ways (8ixig Scooran;), we shall find
that it IS bounded by lines as to length and breadth ; and if we consider the
spherical surface as possessing a form of its own and invested with a fresh
quality, we must regard it as having fitted end on to beginning and made
the two ends (or extremities) one, t^ing thus one potentially only, and not in
actuality." ^

Definition 7. ^.^JVoW-^-- '^' ^ *"

*EiriircSoc hrij^mftijL iarw, igfrif i( laov reus ^^* iavrrji cv^ciacf Kcirat.
A plane surfieice is a surface which lies evenly with the straight lines on
itself

The Greek follows exactly the definition of a straight line mutatis mutandis^
i.e.' with raif...cvdffiaif for Tor«...cn7/xciOi«. Proclus remarks that, in general,
all the definitions of a straight fa'ne can be adapted to the plane surface by
merely changing the ^nus. Thus, for instance, a plane surface is " a surface
the middle of which covers the ends " (this being the adaptation of Plato's
definition of a straight line). Whether Plato actually gave this as the defini-
tion of a plane surface or not, I believe that Euclid's definition of a plane
surface as lying evenly with the straight litus on itself was intended simply to
express the same idea without any implied appeal to vision (just as in the
corresponding case of the definition of a straight line).

I As already noted under Def. 4, Proclus tries to read into Euclid's defini-
tion the Archimedeao assumption that "of surfaces which have the same
extremities, if those extremities are in a plane, the plane is the least" But,

' as I have stated, his interpretation of the words seems impossible, although it

I is adopted by Simplidus also (see an-Nairizi).

I Anjcient alternatives.

The other ancient definitions recorded are as follows.
I I. The surface which is stretched to the utmost {iw oKpov rcro/icn;) : a
1 definition which Proclus describes as equivalent to Euclid's definition ([on
j Proclus' own view of that definition). Cf. Heron, Def. 11, "(a surface) which

is right (and) stretched out " (opOi^ cSa-a dwoT€TafUinj\ words which he adds to

Eudbid's definition.

1 73 BOOK I [l DBF. 7

2. 71k€ least surface among ail thast which have the same exiremiHet.
Proclus is here (p. 1 17, 9) obviously quoting the Axchiaiedean assumfUom.

3. A surface all the parts of which have the property ofJUtmg am {each
other) (Heron, Def. 11).

4. A surface such thai a straight Ume JUs on ail parts of it (Proclosi
p. 117, 8), or such that the UraighiHne fits on iialiways^ i.e. however pboed
(Proclus, p. 117, 20).

With diis should be compared :

5. "(^ plane surface is) such thai^ if a straight line peus through two
points on it, the line coincides wholly with it at every spot^ all ways/* ie. however
placed (one way or the reverse, no matter how), 1^ ^rctJoy tvo ai^iMim Sifnim
cvtfffui, jcol oktf avn; icara wearra rovor wamitK i^apfiiCenu, (Heron, Defl il).
This appears, with the words mtra wivra riww wrolm omitted, in Theon of
Smyrna (p. 112, 5, ed. HillerX so that it goes back at least as fiur as the
ist c A.D. It is of course the same as the dfefinition qommonly attributed to
Robert Simson, and very widely adopted as a substitute for Euclid's.

This same definition appears also in an-Nairizi (ed. Curtse, p. zo) who,
after quoting Simplicius' explanation (on the same lines as Proclus') of the
meaning of Euclid's definition, goes on to say that ''others defined the plane
surface as that in which it is possible to draw a straight line from any pdnt !
to any other." ♦

DiflBculties in ordinary definitions.

Gauss observed in a letter to Bessel that the definition of a plane surfiioe
as a surface such that^ if any two points in it ie tahen^ the straight line joining
them lies wholly in the surface (which, for short, we will call ''Simson*^"
definition) contains more than is necessary, in that a plane can be obtained by
simply projecting a straight line lying in it from a point outside the line but also
lying on the plane ; in fact the definition includes a theorem, or postulate, as
well. The same is true of Euclid's definition of a plane as the surface which
" lies evenly with (all) the straight lines on itself^" because it is suffident for a
definition of a plane if the surface *' lies evenly " with those lines onl^ which
pass through a fixed point on it and each of the several points of a straight line
also lying m it but not passing through the point But from Euclid's point
of view it is immaterial whether a definition contains more than the necessary
minimum provided that the existence of a thing possessing all the attributes
contained in the definition is afterwards proved. This however is not done
in regard to the plane. No proposition about the nature of a plane as such
appears before Book xi., .although its existence is presupposed in all the
geometrical Books i. — iv. and vi. ; nor in Book xi. is there any attempt to
prove, e.g. by construction, the existence of a surface conforming to the
definition. The explanation may be that the existence of the plane as defined
was deliberately assumed from the beginning like that of points and lines, the
existence of which, according to Anstotle, must be assumed as principles
unproved, while the existence of everything else must be proved ; and it may
well«be that Aristotle would have included plane surfaces with points and
lines in this statement had it not been that he generally took his illustrations
horn plane geometry (excluding solid).

But, whatever definition of a plane is taken, the evolution of its essential
properties is extraordinarily difficult. Crelle, who wrote an elaborate article
Zur Theorie der Ebene (read in the Academic der Wissenschaften in 1834) of
which account must be taken in any full history of the subject, observes that,

I. DEF. 7] NOTE ON DEFINITION 7 173

since the plane is the field, as it were, of almost all the rest of geometry, while
a proper conception of it is necessary to enable Eucl. i. i to be understood,
it might have been expected that the theory of the plane would have been the
subject of at least the same amount of attention as, say, that of parallels. This
however was far from being the case, perhaps because the subject of parallels
(which, for the rest, presuppose the notion of a plane) is much easier than that
of the plane. The nature of the difficulties as regards the plane have also
been pointed out recently by Mr Frankland {The First Booh of Euclid's
Elements^ Cambridge, 1905): it would appear that, whatever definition is
taken, whether the simplest (as containing the minimum necessary to deter-
mine a plane) or the more complex, e.g. Simson's, some postulate has to be
assumed in addition before the fundamental properties, or the truth of the
other definitions, can be established. Crelle notes the same thing as r^ards
Simson's definition, containing more than is necessary. Suppose a plane in
which lies the triangle ABC. Let AD join the vertex A

^ to any point D on BC^ and BE the vertex B to any

J point E on CA. Then, according to the definition, AD
lies wholly in the plane of the triangle; so does BE.
But, if both AD and BE are to lie wholly in the one
plane, AD^ BE must intersect, say at F\ if they did not,
there would be two planes in question, not one. But the fact that the lines
intersect and that, say, AD does not pass above or below BE^ is by no
means self-evident.

Mr Frankland points out the similar difficulty as regards the simpler
definition of a plane as the surface generated by a straight
line passing always through a fixed point and always
intersecting a fixed straight line. Let OPF^ OQQ
drawn from O intersect the straight line X 2X P^ Q
respectively. Let R be any third point on X\ then it
^ needs to be proved that OR intersects P'Q in some ^ ^\

point, say R*. Without some postulate, however, it is
not easy to see how to prove this, or even to prove that P'Q intersects X.

Crelle's essay. Definitions by Fourier, Deahna, Becker.

Crelle takes as the standard of a good definition that it shall be, not only as
simple as possible, but also the best adapted for deducing, with the aid of the
simplest possible principles, further properties belonging to the thing defined.
He was much attracted by a very lucid definition, due, he says, to Fourier,
according to which a platu is formed by the aggregate of ail the straight lines
which^ passing through one point on a straight line in space^ are perpendicular
to that straight line. (This is really no more than an adaptation from Euclid's
proposition xi. 5, to the efiect tha^ if one of four concurrent straight lines be
at right angles to each of the other three, those three are in one plane, which
proposition is also used in Aristotle, Meteorologica in. 3, 373 a 13.) But
Crelle confesses that he had not been able to deduce the necessary properties
from this and had had to substitute the definition, already mentioned, of a
plane as the surface containing^ throughout their whole lengthy all the straight
lines passing through a fioud point and also intersecting a straight line in space \
and he only claims to have proved, after a long series of propositions, that the
"Fourier"- or " perpendicular "-surface and the plane of the other definition
just given ate identical, after which the properties of the " Fourier "-surface
can be used along with those of the plane. The advantage of the Fourier
definition is that it leads easily, by means of the two propositions that

174 BOOK I [l

triangles are equal in all respects (i) when two sides and the induded an^
are respectively equal and (2) when all three sides are respectively equal, to the
property expressed in Siroson's definition. But Crelle uses to establish these
two congruence-theorems a number of propositions about equal angles^ si^k-
meniary angles, right angles, greater and Use angles ; and it is difficult to
question the soundness of Schotten's criticisin that these notions in themselves
really presuppose that of a plane. The difficulty due to Fourier's use of
the word '' perpendicular,*' if that were all, could no doubt be got over. Thus
Deabna in a dissertation (Marburg, 1837) constructed a plane as follows.
Presupposing the notions of a straight line and a qphere, he observes that, if a
sphere revolve about a diameter, all the points of its surftce which move
describe closed curves (circles). Each of mese circles, during the revolution,
moves along itself, and one of them divides the sur&ce of the qpheie into two
congruent parts. The aggrc^te then of the lines joining the centre to the
points of this circle forms ih^ plane. Again, J. K. Becker (Die ElemetUe der
Geometries 1877) pointed out that the revolution of a right ang^e about one
side of it produces a conical suifice which differs from all other conical
surfaces generated by the revolution of oAer angles in the foct that the
particular cone coincides with the cone vertically opposite to ii : this characteristic
might therefore be taken in order to get rid of the use of the rig^ angfe.

W. Bolyai and Lobachewsky.

Very similar to Deahna's equivalent for Fourier's definition is the device
of W. Bolyai and LobachewBky (described by Frischauf, Elemente ier
absoluten Geometries 1876). They worked upon a fundamental idea first
suggested, apparently, by Leibniz. Briefly stated, their wi^ of evolving a
plane and a straight line was as follows. Conceive an infinite number of
pairs of concentric spheres described about two fixed points in space, (7, <7,
as centres, and with equal radii, gradually increasing : these pairs of equal
spherical surfaces intersect respectively in homogeneous curves (circles), and >
the '* Inbegriff *' or aggr^ate of these curves of intersection forms a plane.
If ^ be a point on one of these circles {k say), suppose points 3f, M' to start
simultaneously from A and to move in opposite directions at the same speed
till they meet at BsSBLy, B then is *' opposite" to A^ and A^ B divide the
circumference into two equal halves. If the points ^, ^ be held fast and the
whole system be turned about them until O takes the place of 0^ and O of
Os the circle k will occupy the same position as before (though turned a
different way). Two opposite points, P^ Q say, of each of the other circles
will remain stationary during the motion as well as A, B: the " Inbegriff " or
aggregate of all such points which remain stationary forms a straight line. It
is next observed that the plane as defined can be generated by the revolution
of the straight line about Off^ and this suggests the following construction
for a plane. Let a circle as one of the curves of intersection of the pairs of
spherical surfaces be divided as before into two equal halves at A^ B. Let the
arc ADB be similarly bisected at Z>, and let C be the
middle point of AB. This determines a straight line CD
which is then defined as ''perpendicular" to AB. The revo-
lution of CD about AB generates a plane. The property
stated in Simson's definition is then proved by means of the
congruence-theorems proved in Eucl. i. 8 and i. 4. The
first is taken as proved, practically by considerations of
symmetry and homogeneity. If two spherical surfaces, not necessarily equal,
with centres O^ O intersHSCt, A and its ''opposite" point B are taken as

\ L DEF. 7]

NOTE ON DEFINITION 7

175

before on the curve of intersection (a circle) and, relatively to Off^ the point
A is taken to be convertible with B or any other point on the homogeneous
curve. The second (that of Eucl. i. 4) is established by simple application.
Rausenberger objects to these proofs on the 'grounds that the first assumes
that the two spherical surfaces intersect in one single curve, not in several,
and that the second compares ang/es : a comparison which, he says, is possible
only in a p/ant^ so that a plane is really presupposed. Perhaps as regards
the particular comparison of angles Rausenberger is hypercritical; but it is
difficult to r^ard the supposed proof of the theorem of Eucl. i. 8 as sufficiently
rigorous (quite apart from the use of the uniform mo/ion of points for the
purpose of bisecting lines).

Simson's property is proved from the two congruence-theorems thus.
Suppose that AB is *' perpendicular" (as defined by Bolyai) to two generators
CJff CN of a plane, or suppose CM^ CN respectively to make with AB two
angles congruent with one another. It is enough to prove that, if Z' be any
pomt on the straight line MN^ then CP^ just as
much as CM^ CW respectively, makes with AB two
angles congruent with one another and is therefore
a generator. We prove successively the congruence
of the following pairs of triangles :

ACM, BCM

ACN, BCN

AMN, BMN

AMP, BMP

ACP, BCP,
whence the angles ACP, BCPzx^ congruent
Other views.

Enriques and Amaldi (EUmenti di geometria, Bologna, 1905), Veronese
(in his EUmenti) and Hilbert all assume as a postulate the property stated in
Simson's definition. But G. Ingrami {Elementi di geometria, Bologna, 1904)
proves it in the course of a remarkable series of closely argued proposition
oased upon a much less comprehensive postulate. He evolves the theory of
the plane from that of a triangle, beginning with a triangle as a mere three-side
(trilatero), i.e. a frame, as it were. His postulate relates to the three-side and
is to the effect that each " (rectilineal) segment " joining a vertex to a point of
the opposite side meets every segment similarly joining each of the other two
vertices to the points of the sides opposite to them respectively, and, con-
versely, if a point be taken on a segment joining a vertex to a point of the
opposite side, and if a straight line be drawn from another vertex to the point
on the segment so taken, it will if produced meet the opposite side. A
triangle is then defined as the figure formed by the aggregate of all the
segments joining the respective vertices of a thru-side to points on the
opposite sides. After a series of propositions, Ingrami evolves a plane as the
figure farmed by the " half straight-lines " which project from an internal point
^ the trian^e the points of the perimeter, and then, after two more theorems,
proves that a plane is determined by any three of its points which are not in
a straight line, and that a straight line which has tu^o points in a plane has all
Us points in it.

The argument by which Bolyai and Lobachewsky evolved the plane is
of course equivalent to the definition of a plane as the locus ef all points
eqmdistant from two fixed points in space.

176 BOOK I [l mrr. 7—9

Leibniz in a letter to Giordano defined a plane as ikai tmtfaa wUdk
divides space into two congruemi parts. Admting to Giordano's cnticisni that
you could conceive of surfaces and lines which divided qwce or a plane into
two congruent parts without beipg/£iff^ or i/rw^A/ respectively. Bees (jOiar
EuJUidische und NichtEuklidiseke Geameirie^ 1888) pointed out that what was
wanted to complete the definition was the further condition that the two
congruent spaces could be sHi tUong each other without the surfiuxs ceasiiy
to coincide, and claimed priority for his completion of the definition in dus
way. But the idea of all the parts of a i^ane fitting enctly on eUl other parts
is ancient, appearing, as we have seen, in Heron, Det ii.

Definitions 8, 9.

8. 'EvfircSoc Sc ywFui coTcv 1} i» hnwAf tuo jpa§t§aAy iano^ubm^ AXXajXmw
fcoi fi^ iw* fMciac K€ifjL€Viiiiy wpit oAX^Xac tmk ypafi§im¥ kkun^

9. ^Orav Sc al xi/Mcxovcnu r^ ytirior yfM^ngud Mumi Sau^, Mvypapt^un
jcoXciTcu 17 yuyta.

8. A plane angle is the inelisuttion to one another of two Hnes m a pkme
which meet one another and do mot lie in a straight Usie.

9. And when the lines containing the angit are straight^ the angle is tailed
rectilineal.

The phrase "not in a straight line** is strange, seeing that the definition
purports to apply to angles formed by curves as well as straight lines. We
should rather have expected eontmnous (om^xf^) with one another; and
Heron takes this to be the meaning, since he at once adds an expknation as
to what is meant by lines not being continuous (oA ovi^xm). It looks as though
Euclid really intended to define a rectilineal angle, but on second thoughts^
as a concession to the then common recognition of curvilineal angles, altered
^ straight lines " into " lines " and separated the definition into two.

I think ril our evidence suggests that Euclid's definition of an angle as
inclination (xXuric) was a new departure. The word does not occur in
Aristotle ; and we should gather fi-om him that the idea generallv associated
with an angle in his time was rather deflection or breahing of lines (xXiurtv) : cf.
his common use of #cc#cAairtfai and other parts of the verb fcXoK, and also his
reference to one bent line forming an angle (r^r KtKOfjL^iiyqy koX ^^oiNroy yvn^iav^
Jfetaph, 1016 a 13).

Proclus has a long and elaborate note on this definition, much of which
(pp. 121, 12 — 126, 6) is apparently taken direct fi-om a work by his master
Syrianus (6 iJ/«^cpos Ka$frf€fjMv). Two criticisms contained in the note need
occasion no difficulty. One of these asks how, if an angle be an inclination,
one inclination can produce two angles. The other (p. 128, 2) is to the effect
that the definition seems to exclude an angle formed by one and the same
curve with itself, e.g. the complete a'ssoid [at what we call the '^ cusp "1 or the
curve known as the hippopede (horse-fetter) [shaped like a lemniscatej. But i
such an "angle" as this belongs to higher geometry, which Euclid may well
be excused for leaving out of account in any case.

Other ancient definitions : ApoUonius, Plutarch, Carpus.

Proclus* note records other definitions of great interest ApoUonius
defined an angle as a contracting of a surface or a solid at one Ooint under a
hrohen line or surface (onnayory^ crt^oyciav ^ oripcov 9po% l¥i oiffMcIy vwh
K€KXaafUtqg ypo/^v ^ ^t^omigi), where again an angle is supposed to be
formed by one broken line or surface. Still more interesting, perhaps, is the
definition by "those who say that the first distance under the point (to vywrov

I

I. Dkff. 8, 9] NOTES ON DEFINITIONS 7—9 177

iiaarrifia vtrb ro ainUiov) is the angle. Among these is Plutarch, who insists
that Apollonius meant the same thing ; for, he says, there must be some first
distance under the breaking (or deflection^ of the including lines or surfaces,
though, the distance under the point bemg continuous, it is impossible to
obtain the actual firsty since every distance is divisible without limit " (cir*
axc(f>ov). There is some vagueness in the use of the word " distance" {^^Mjtm\^ \
thus it was objected that ** if we anyhow separate off the/rj/" (distance being
apparently the word understood) ''and draw a straight Ime through it^ we get
a triangle and not one angle.'* In spite of the objection, I cannot but see in
the idea of Plutarch and the others the germ of a valuable conception in
infinitesimals, an attempt (though partial and imperfect) to get at the rate
of divergence between the lines at their point of meeting as a measure of the
angle between them.

A third view of an angle was that of Carpus of Andoch, who said " that
the angle was a quantity (itoctof), namely a distana {Stdarrifia) between the
lines or surfaces containing it. This means that it would be a distance (or
divergence) in one sense (c^* ty Sccorctfc), although the angle is not on that
account a straight line. For it is not everything extended in one sense (to ^^' tv
ScaoraroV) that is a line." This very phrase " extended one way" being held
to define a line^ it is natural that Carpus' idea should have been described as
the greatest possible paradox (Trarroii^ irapaSo^orarov). The difficulty seems to
have been caused by the want of a different technical term to express a new
idea ; for Carpus seems undoubtedly to have been anticipating the more
modem idea of an angle as representing divergence rather than distance, and to
have meant by c^' tv in one sense (rotationally) as distinct from one way or in
one dimension (linearly).

To what category does an angle belong?

There was much debate among philosophers as to the particular category
(according to the Aristotelian scheme) in which an angle should be placed ;
is it, namely, a quantum (^fifT6v\ quale (iroiov) or relation (n-pos n) ?

1. Those who put it in the category o{ quantity argued from the fact that
a plane angle is divided by a line and a solid angle by a surface. Since, then,
it is a surface which is divided by a line, and a solid which is divided by
a surface, they felt obliged to conclude that an angle is a surface or a solid, and
therefore a magnitude. But homogeneous finite magnitudes, e.g. plane
angles, must bear a ratio to one another, or one must be capable of being
multiplied until it exceeds the other. This is, however, not the case with a
rectilineal angle and the horn-like angle (iccparofi^'s), by which latter is meant
the "angle" between a circle and a tangent to it, since (Eucl. in. 16) the
latter "angle" is less than any rectilineal angle whatever. The objection, it
will be observed, assumes that the two sorts of angles are homogeneous.
Plutarch and Carpus are classed among those who, in one way or other, placed
an angle among magnitudes-^ and, as above noted, Plutarch claimed Apollonius
as a supporter of his view, although the word contraction \o{ a surface or solid)

I used by the latter does not in itself suggest magnitude much more than Euclid's
I inclination. It was this last consideration which doubtless led " Aganis," the
; " friend " (socius) apparently of Simplicius, to substitute for Apollonius'
j wording " a quantity which has dimensions and the extremities of which arrive
at one point'' (an-Nairizi, p. 13).

2. Eudemus the Peripatetic, who wrote a whole work on the angle, main-
tained that it belonged to the category of quality. Aristotle had given as his
fourth variety of quality "figure and the shape subsisting in each thincj. and,

H. £. 12

I

17^ BOOK I [l Dcfp. 8, 9

besides these, straightness, curvature, and the like** (CMig^ries 8» 10 a 11).
He says that each individual thing is spoken of as fuaie in respect of its fbmiy
and he instances a triangle and a square, using them again later on (ifiU, 1 1 a 5)
to show that it is not all qualities which are susceptible of man and less ; again,
in Physics i. 5,, 188 a 25 a«^, strai^^ drcmlar are called kinds fA fy^re.
Aristotle would no doubt have regarded deJUciwn (ffdcXdEirtfai) as belonging to
the same category with straightness and curvature (fcofuraXoniv). At all eventi,
Eudemus took up an angle as having its origin in the hnakmg or i^bdhn
(fcAocrcc) of lines : deflection, he aigued, was quality if straightness was, and that
which has its origin in quality is itself quality. Objectors to this view argued
thus. If an an^le be a quality Qrmanyc) like heat or cold, how can it be bisCMSted,
say ? It can m fact be divided ; and, if things of which divisibility is an
essential attribute are varieties of quantum and not qualities, an angle cannot
be a quality. Further, the man and the kss are the appropriate attributes tA
quality, not the equal and the unequal ; if therefore an ande were a quality,
we should have to say of angles, not that one is greater and another smaller,
but that one is more an angle and another less an ang^ and that two angles
are not unequal but dissimilar (nSrofUHOi). As a matter of fiurt, we are tokf by
Simplicius, 538, 21, on Arist Da eaala that those who brought the an(^ uiKter
the category of quale did call equal angles similar angles ; and Aristotle
himself speaks of similar angles in this sense in Da coda 296 b 20, 311 b 34.

3. Euclid and all who cidled an angle an inclination are held by Sjrrianos
to have classed it as a nlation (vpof rt). Yet Euclid certainly rqparded varies
as magnitudes; this is clear both -from the earliest propositions deaSng
specifically with angles, e.g. i. 9, 13, and also ^though in another way) from
his describing an angle in the very next defimtion and always as camAumed
(n€pi€xofuyri) by the two lines forming it (Simon, Euclid^ p. 28).

Proclus (i.e. in this case Syrianus) adds that the truth lies between these
three views. The angle partakes in fact of all those categories: it needs the
quantity involved in magnitude, thereby becoming susceptible of equality,
inequality and the like ; it needs the quality given it by its form^ and lastly
the nlation subsisting between the lines or planes bounding it

Ancient classification of '* angles."

An elaborate classification of angles given by Proclus (pp. 126, 7 — 127, 16)
may safely be attributed to (jeminus. In order to show it by a diagram it I

Angles

on sarfaces in solids
I {iv OTcpewt)

r-

on simpU surfaces on mixed surfaces

I (e.g. cones, cylinders)

I ' 1

on phtus on spherical surfaces

made by ntuple lines made by **mixid*^ lines bjr one of each

(e.g. the angle made by a (e.g. the angle forAied by an

curve, such as the cissM ellipse and its axis or by

and kippapede, with itself) an ellipse and a drde)

line-line line-drcumf. drcumf.-circumf.

1 1 1 . H ,,

Ime-convcx Ime-concave convex-convex concave-concave ikiued, or

(eg. an^e of a e.g. horn-like (dfu^vproi) {dft^oiKM) convex-concave

semicircle) {xtparoti^t) or ••scraper-like" (e.g. those of

((MTporcdfcf) Utnes)

1. Deff. 8, 9] NOTE. ON DEFINITIONS 8, 9 179

will be necessary to make a convention about terms. Angles are to be under-
stood under each class, " line-circumference *' means an angle contained by a
straight line and an arc of a circle, " line-convex " an angle contained by a
straight line and a circular arc with convexity outwards^ and so on in every
case.

Definitions of angle classified.

As for the point, straight line, and plane, so for the angU^ Schotten gives
a valuable summary, classification and criticism of the different modem views
up to date (Inhalt und Methode des planimetrischen Unterrichts^ 11., 1893,
pp. 94 — 183); and for later developments represented by Veronese reference
I may be made to the second article (by Amaldi) in Questioni riguardanti
\ la geometria eUmentare (Bologna, 1900) already referred to.
I With one or two exceptions, says Schotten, the definitions of an angle may

f be classed in three groups representing generally the following views :
\ ' I. The angle is the difference of direction between tUH> straight lines. (With
JL this group may be compared Euclid's definition of an angle as an inclination.)
I 2. The angle is the quantity or amount (or the measure) of the rotation
\ necessary to bring one of its sides from its own position to that of the other side
without its moving out of the plane containing both,

3. The angle is the 'portion of a plane included between ttvo straight lines in
the plane which meet in a point {or two rays issuing from the point).

It is remarkable however that nearly all of the text-books which give
definitions different from those in group 2 add to them something pointing to
a connexion between an angle and rotation : a striking indication that the
essential nature of an angle is closely connected with rotation, and that a good
definition must take account of that connexion.

The definitions in the first group must be admitted to be tautologous, or

circular^ inasmuch as they really presuppose some conception of an angle.

Direction (as between two given points) may no doubt be regarded as a primary

k notion; and it may be defined as "the immediate relation of two points which

} the ray enables us to realise'' (Schotten). But ''a direction is no intensive

^ magnitude, and therefore two directions cannot have any quantitative

I difference" (Biirklen). Nor is direction susceptible of differences such as

'those between qualities, e.g. colours. Direction is a jiVi^ar' entity: there

cannot be different sorts or degrees of direction. If we speak of "a different

direction," we use the word equivocally ; what we mean is simply "another"

direction. The fact is that these definitions of an angle as a difference of

direction unconsciously appeal to something outside the notion of direction

. altogether, to some conception equivalent to that of the angle itself.

Recent Italian views.

I The second group of definitions are (says Amaldi) based on the idea of the
rotation of a straight line or ray in a plane about a point : an idea which,
I logically formulated, may lead to a convenient method of introducing the
I angle. But it must be made independent of metric conceptions, or of the
conception of congruence^ so as to bring out first the notion of an angle, and
afterwards the notion of ^^wa/ angles.

The third group of definitions satisfy the condition of not including metric
conceptions ; but they do not entirely correspond to our intuitive conception
of an angle, to which we attribute the character of an entity in one dimension
i(as Veronese says) with respect to the ray as element, or an entity in tivo

12-

iSo BOOK I [l Dsr. 9

dimensions with reference Xo points as elementSi which may be called an attgtdar
sector. The defect is however easily remedied by considering the an^ as
" the aggr^te of the rays issuing from the vertex and comprised in the angular .
sector."

Proceeding to consider the principal methods of arriving at the logical
formulation of the first superficial properties of the ftam from whidi a
definition of the angle may emerge, Amaldi distinguishes two points of view
(1) XhtgtfutiCy (2) the actual.

(i) From the first point of view we consider the dmster of straight Mnes
or rays (the aggregate of all the straight lines in a plane oassing throush a
point, or of all the rays with their extremities in that point) as generatea by
the movement of a straight line or ray in the plane, about a point This leads
to the postulation of a closed order^ or circular disposition^ of the straight lines
or rays in a cluster. Next comes the connodon subsisting between the
disposition of any two clusters whatever in one plane, and so oa

(2) Starting from the point of view of the aitual^ we lay the foundation
of the definition of an angle in the dimmn of the plane into two parts pialf-
planes) by the straight line. Next, two straight lines (tf, i) in the plane, inter-
secting at a point O^ divide the plane into four regions which are called
angular sectors (convex) ; and finally the an^e (ab) or (Ai) may be defined as
the aggregate of the rays issuing firam O CMd belonging to the angular sector
which has a and hfor sides.

Veronese's procedure (in his Elementi) is as follows. He bq;ins with the
first properties of the plane introduced by^ the following definition.

The figure given by all l^e straight lines joining Uie pomts of a straight
line r to a point /' outside it and by
the parallel to r through /' is called a
cluster of straight lineSy a cluster of rays^
or a plane^ according as we consider

the element of the figure itself to be the ^^y"^ j

straight line^ the ray terminated at F^ P ^

or z, point. I

[It will be observed that this method of producing a plane involves using {
the parallel to r. This presents no difficulty to Veronese because he has 4
previously defined parallels, without reference to the plane, by means of reflex
or opposite figures, with respect to a point O : '' two straight lines are csdled <
parallel^ if one of them contains two points opposite to (or the reflex of) two
points of the other with respect to the middle point of a common transversal
(of the two lines)." He proves by means of a postulate that the parallel r
does belong to the plane Pr. Ingrami avoids, the use of the parallel by
defining a plane as "the figure formed by the half straight lines which project (
'fi-om an internal point of a triangle (i.e. a point on a line joining any vertex of
a three-side to a point of the opposite side) the points of its perimeter," and "
then defining a cluster of rays as "the aggregate of the half straight lines in a
plane ^starting from a given point of the plane and passing through the points I
of the perimeter of a triangle containing the point."] J

Veronese goes on to the definition of an angle. " We call an angle apart
of a duster o/rays^ bounded by two rays (as the segment is a part of a straight '
line bounded by two points).

^*An angle of the cluster^ the bounding rays of which are opposite^ is caUod a
flat angle."

Then, after a postulate corresponding to postulates which he lays down for

I. Deff. 9-12] NOTES ON DEFINITIONS 9—12 181

a rectilineal segment and for a straight line^ Veronese proves that all fiat angles
are equal to one another.

Uf9

Hence he concludes that "the duster of rays is a homogeneous linear
system in which the element is the ray instead of the point. The cluster
being a homogeneous linear system, all the propositions deduced from
[Veronese's] Post i for the straight line apply to it, e.g. that relative to
the sum and difference of the segments : it is only necessary to substitute
the ray for the point, and the angle for the segment.*'

Definitions 10, 11, 12.

10. *Orav 8c cMciOi lit cMcIaF orrotfcura ras ^^€^79 ytavta.% uras liXXi/Xaiv
ir<k|J, ^ptfi; kKuripa toiv Tcraiv fa¥%mv inrif koI 17 ^^con^jcvia cMcui icd^croc fcoAcirai,

1 1. *Afi)8Xcia yiavia iariv 1} /ittltop 6pBrj^

12. *Ofcui Sk ff IKaaatav 6p$7Js.

10. fVhen a straight line set up on a straight line makes the adjacent angles
equal to one anothery each of the equal angles is right, and the straight line
stcmding on the other is called a perpendicular to that on which it stands,

I II. An obtuse angle is an angle greater than a right angle.

\ 1 2. An acute angle is an angle less than a right angle,

\ ^^€^ is the regular term for adjacent angles, meaning literally " (next) in

I ' order.'' . I do not find the term used in Aristotle of angles^ but he explains its
i meaning in such passages as Physics vi. i, 231 b 8 : "those things are (next)
i in order which have nothing of the same kind {crvyytvk) between them."
► fca^crof , perpendicular^ means literally let fall : the full expression is perpen-

I dicular straight line, as we see from the enunciation of Eucl. i. 11, and the
\ notion is that of a straight line let fall upon the surface of the earthy dLpiumb-
[ line, Proclus (p. 283, 9) tells us that in ancient times the perpendicular was
called gnomon-tvise (#cara yv«ifioFa), because the gnomon (an upright stick) wtt^
f set up at right angles to the horizon.

The three kinds of angles are among the things which according to the
Platonic Socrates {Republic vi. 510 c) the geometer assumes and argues from,
declining to give any account of them because they are obvious. Aristotle
discusses the priority of the right angle in comparison with the acute (Metaph.
1084 b 7): in one way the right angle is prior, i.e. in being defined (^n
Zpurraj) and by its notion (rf Xoy<{»), in another way the acute is prior, i.e. as

I being a party and because the right angle is divided into acute angles ; the
acute angle is prior as mattery the right angle in respect oi form\ cf. also
Metaph, 1035 b 6, ''the notion of the right angle is not divided into

Definition 14.

182 BOOK I [i. Deff. 12-14

that of an acute angle, but the reverse ; for, when defining an acute angle,
you make use of the right angle.*' Proclus (p. 133, 15) observes that it is by
the perpendicular that we measure the heights of figures, and that it is by
reference to the right angle that we distinguish the other rectilineal angles,
which are otherwise undistinguished the one from the other.

The Aristotelian Problems (16, 4, 913 b ^6) contain an ex|xression perhaps
worth quoting. The question discussed is why things which &11 on the
ground and rebound mikt ''similar" an^es with thMuraice on both sides of |
the point of impact; and it is observed that '^.die right angle is the Nmii ^
(ofMs) of the opposite angles," where however *' opposite " seems to mean, not
''supplementary " (or acute and obtuse), but the equal angles made with the ^
surface on opposite sides of the perpendicular.

Proclus, after his manner, remarks that the statement that an angle less
than a right angle is acute is not true without qualification, for f i]| the kmihUke
angle (between the circumference of a drde ai\d a tangent) is less than a
right angle, since it is less than an atuU angle, but is not an acute angle, wbik
(2) the "angle of a semicircle " (between the arc and a diameter) is also less
than a right angle,* but is not an acute angle. %

The existence of the right angle is X)f course proved in L 1 1.

Definition 13.

Opo% iariv, o tivos iari wipa$.

A boundary is that which is an extremity rfemy thing.

Aristotle also uses the words 2pof and t^s as synonymous. Cf. Drgm,
animaL ii. 6, 745 a 6, 9, where in the expression "h'mit of magnitude** first
one and then the other word is used.

Proclus (p. 136, 8) remarks that the word boundary is appropriate to the
origin of geometry, which began from the measurement of areas of ground
and involved the marking of .boundaries.

i

Sx^fui loTi TO viro TIV09 17 rivwv op«iiv mpuxofityov, .

A figure is that rvhich is contained by any boundary or boundaries. \

Plato in the Meno observes that roundness (trrpor^Xifnfi) or the round is a ^

" figufe," and that the straight and many other thmgs are so too ; he then j

inquires what there is common to all of thejxi, in virtue of which we apply the |

term "figure" to them. His answer is (76 a): "with reference to every j
fi^re I say that that in "u^hich the solid terminates (rovro^ ci9 u rh oryMlw

irtfHuv€C) is a figure^ or, to put it briefly, a figure is an extremity of a solid.** I
The first observation is similar to Aristotle's in the Physics i. 5, 188 a 25, <
where angle, straight, and circular are mentioned as genera of figure: In the
Categories 8, 10 a 11, "figure" is placed with straighthess and curvedness in
the category' of quality. Here however "figure" appears to mean skafe I
(fiap^') rather than " figure " in oiir sense. Coming hearer to "figure" in our !
sense, Aristotle admits that figure is ""a sort of magnitude" i^De anima in. i, \
425 a 18X and he distinguishes plcme figures of two kinds, in language not
unlike Euclid's, as contained by straight and circular lines re^)ectively : "every
plane figure is either rectilined or formed by circular- lines (vcpt^cpoypa/ifuiy),
and the rectilineal figure is contained by several lines, the circular by one
line " {De caelo 11. 4, 286 b 13). He is careful to explain that a plane is not a

I. Deff. 14-16] NOTES ON DEFINITIONS 12—16 183

figure, nor a figure a plane, but that a plane figure constitutes one notion and
is a species of the genus figure {AnaL post, n. 3, 90 b 37). Aristotle does not
attempt to define figure in general, in fact he says it would be useless : " From
this it is clear that there is one definition of soul in the same way as there is
one definition oi figure \ for in the one case there is no figure except the
triangle, quadrilateral, and so on, nor is there any soul other than those above
mentioned. A definition might be constructed which should apply to all
figures but not specially to any particular figure, and similarly with the
species of soul referred to. [But such a general definition would serve no
purpose.1 Hence it is absurd here as elsewhere to seek a general definition
which will not be properly a definition of anything in existence and will not
be applicable to the particular irreducible species before us, to the neglect of
the definition which is so applicable" {De anima 11. 3, 414 b 20^ — 28).

Comparing Euclid's definition with the above, we observe that by. intro-
ducing boundary {opos) he. at once excludes the straight which Aristotle classed
as figure ; he doubtless excluded angle also, as we may judge by (i) Heron's
statement that ''neither one nor two straight lines can complete a figure,"
(3) the alternative definition of a straight line as *'that which cannot with
another line of the same species form a figure," (3) Geminus' distinction
between the line irhxcYi forms a figure (crxfffJLarowoiovaa) and the line which
extends indefinitely {iw* aTrcipov iKpaXXxt/uvri), which latter term includes a
hjrperbola and a parabola. Instead of calling figure an extremity as
Plato did in the expression '^extremity (or limit) of a solid,** Euclid
describes a figure as that which has a boundary or boundaries. And lastly,
in spite of Aristotle's objection, ht does attempt a general definition to
cover all kinds of figure, solid and plane. It appears certain therefore that
Euclid's definition is entirely his own.

Another view of a figure, recalling that of Plato in Meno 76 a, is attributed
by Proclus (p. 143, 8) to Posidonius. The latter regarded the.^^r^ as the
confining extremity or limit (wipas cnrvicXciov), "separating the notion of figure
firom quantity (or magnitude) and making it the cause of definition^ limitation^
and inclusion (roO iipL<r$a%, icoi ir€W€pcur6<u ical t^ itc^mox^s)... Posidonius thus
seems to have in view only the boundary placed round from outside, Euclid
the whole content, so that Euclid will speak of the circle as a figure in
respect of its whole plane (surface) and of its inclusion (firom) without, whereas
Posidonius (makes it a figure) in respect of its circumference... Posidpnius
wished to explain the notion of figure as itself limiting and confining magnitude."

Proclus observes that a logical and refining critic might object to Euclid's

definition as defining the genus from the species, since that which is enclosed

by one boundary and that which * is enclosed by several are both species of

figure. The best answer to this seems to be supplied by the passage of

^ Aristotle's I>e anima quoted above.

; Definitions ,15, 16. ^

I 15. KvfcXos iari <r}(rjfta ciriircSov vwo fiuis ypofifi^c mpuxofifvoy [^ fcaXcinu

T^H^M^>cta|, wp^ ^¥ d^* M^ fnffJL€iov rwv cvros tov fr^mpxiro^ icccficvaiv n-curai at
\ wpovwiirrownii cii^cuu {irpof rrjv rov kvkXov ir€pt/^€p€iay\ urcu dXAifXacc curiv.
' 1 6. Kivrpw 8i rov.KVicXov ro airfp,€iO¥ fcaXcirac

1$. A circle is a J>lane figure contained by one line such that all the straight
lines falling upon it from one point among those lying within the figure are equal
to one another ;

16. And the point is called the ctnXxe of the circle.

i84 BOOK I [i. Dbff. t5» i6

The words ^ KoXctTai xcpc^^io, *' which is called the drcumferenoe," and
wpoi rifv Tov kvkXov w€piitiip€ua»f "to the circuinference of the drde," aie
bracketed by Heiberg because, although the Mss. have them, they aie
omitted in other ancient sources, viz. Prmdus, Taurus, Sextus Empiricus and
Boethius, and Heron also omits the second gloss. The recently discovered
papyrus Herculanensis No. 1061 also quotes the definition without the words
m question, confirming Heibeig's rejection of them (see Heiberg in Hermes
XXXVIII., 1903, p. 47). The words were doubtless added in view of the
occurrence of the word "circumference'' in Deff. 17, 18 immediatdv
following, without any explanation. But no explanation was needed. Though
the word xc/M^cpcia does not occur in Plato, Aristotle uses it seyeml timet

( 1 ) in the general sense of contour without any special mathematical signification,

(2) mathematically, with reference to the rainbow and the circumference, as
well as an arc, of a circle. Hence Euclid was perfectly justified in emtdoying
the word in Deff. 17, 18 and elsewhere, but leaiong it undefined as being a
word universally understood and not involving in itself any mathematical
conception. It may be added that an-NairlzI had not the bracketed words
in his text ; for he comments on and tries to explain Euclid's omission to
define the circumference.

The definition itself contained nothing new in substance. Plato {Parme-
nides 137 e) says : " Round is, I take it, that the extremes of which are every
way equally distant from the middle " (irrpoyyvAor yc vov Icrrt ToOr«s oS & td
laxora warraxj awo rav ficcrov lirov Awixff). In Aristotle we find the following
expressions: "the circular (xfpi^€pdypo/4fu>v) plane figure bounded by one
line" (Df caelo 11. 4, 286 b 13 — 16); "the plane equal (Le. extending equally
all ways) from the middle" (IrArcSoK to lie rov /mctov \ffw\ meaning a
circle (^Rhetoric iii. 6, 1407 b 37); he also contrasts with the circle ^'any
other figure which has not the lines from the middle equal, as for example an
egg-shaped figure" {De caelo ii. 4, 287 a 19). The word "centre" («€»^pw)
was also regularly used : cf. Proclus* quotation from the "oracles" (Aoyia),
" the centre from which all (lines extending) as far as the rim are equal."

The definition as it stands has no genetic character. It says nothing as to
the existence or non-existence of the thing defined or as to the method of
constructing it. It simply explains what is meant by the word " circle," and
is a provisional definition which cannot be used until the existence of circles
is proved or assumed. Generally, in such a case, existence is proved by
actual construction; but here the possibility of constructing the circle as
defined, and consequently its existence, are /^j/^/17/^// (Postulate 3). A genetic
definition might state that a circle is the figure described when a straight line,
always remaining in one plane, moves about one extremity as a fixed point
until it returns to its first position (so Heron, Def. 29).

Simplicius indeed, who points out that the distance between the feet of a
pair of compasses is a straight line from the centre to the circumference, will
have it that Euclid intended by this definition to show how to construct a
circle by the revolution of a straight line about one end as centre ; and an-
Nairizi points to this as the explanation (i) of Euclid's definition of a circle
as 2i plane figure^ meaning the whole surface bounded by the circumference,
and not the circumference itself, and (2) of his omission to mention the
"circumference," since with this construcrion the circumference is not drawn
separately as a line. But it is not necessary to suppose that Eudid himself
did more than follow the traditional view ; for the same conception of the
circle as k plane figure appears, as we have seen, in Aristotle. While, however.

If

I. Deff. 15-17] NOTES ON DEFINITIONS 15—17 185

Euclid is generally careful to say the ^^circumference of a circle " when he means
the circumference, or an arc, only, there are cases where "circle" means
"circumference of a circle," e.g. in in. 10: "A circle does not cut a circle
in more points than two."

Heron, Produs and Simplidus are all careful to point out that the centre
is not the only point which is equidistant from all points of the circumference.
The centre is the only point in the plane of the circle ("lying within the figure,"
as Euclid says) of which this is true; any point not in the same plane which
is equidistant from all points of the circumference is a pote. If you set up a
"gnomon " (an upright stick) at the centre of a circle (i.e. a line through the
centre perpendicular to the plane of the circle), its upper extremity is a pole
(Proclus, p. 153, 3); the perpendicular is the locus of all such poles.

Definition 17.

Ataficr/>09 8c rov kvkXov ktrrXv cMcui ric 81a rw Kiirrpov rjyitiyti ical irtparov-
ficn; i^* hcaripa ra /upij vro r^^ rev kvkXov w€pif^€p€ias, i/ric icoi &\a tc/avci rov

JCVfcAoF.

A diameter of the circle is any straight lifie drawn through the centre and
terminated in both directions by the circumference of the circle^ and such a ^reught
line also bisects the circle.

The last words, literally "which (straight line) also bisects the circle,"
are omitted by Simson and the editors who followed him. But they are
necessary even though they do not "belong to the definition" but only
express a property of the diameter as defined. For, without this explanation,
Euclid would not have been justified in describing as a j^/rrAnrcle a portion
of a circle bounded by a diameter and the circumference cut off by it

Simplicius observes that the diameter is so called because it passes through
the whole surface of a circle as if measuring it, and also because it divides the
circle into two equal paits. He might however have added that, in general, it
is a line passing through a figure where it is widest^ as well as dividing it
equally: thus in Aristotle ra #cara huL/urpoy icct/Acva, "things diametrically
situated ** in space, are at their maximum distance apart Diameter was the
regular word in Euclid and elsewhere for the diameter of a square^ and also
of a parallelogram; diagonal (Siayuvios) was a later term, defined by Heron
(Def. 68) as the straight line drawn from an angle to an angle.

Proclus (p. 157, 10) says that Thales was the first to prove that a circle is
bisected by its diameter; but we are not told how he proved it Proclus gives
as the reason of the property " the undeviating course of the straight line
through the centre " (a simple appeal to symmetry), but adds that, if it is
desired to prove it mathematically, it is only necessary to imagine the diameter
drawn and one part of the circle applied to the other ; it is then clear that
they must coincide, for, if they did not, and one fell inside or outside the
other, the straight lines from the centre to the circumference would not all be
equal : which is absurd.

Saccheri's proof is worth quoting. It depends on three "Lemmas"
immediately preceding, (i) that two straight lines cannot enclose a space,

!2^ that two straight lines cannot have one and the same segment common,
3) that, if two straight lines meet at a point, they do not touch, but cut one
another, at it.

" Let MDHNKM be a circle, A its centre, MN a diameter. Suppose

i

i86 BOOK I [i. Dkff. 17,

the portion MNKM of the circle turned about the fixed points M^ N^ 90
that it ultimately comes near to or coincides with the remaining portion
MNHDM.

"Then (i) the whole diameter MAl^^ with all
its points, clearly remains in the same position,
since otherwise two straight lines would enclose a
space (contrary to the first Lemma).

'' (ii) Clearly no point K of the circumference
NKM falls withm or outside the surfieure enclosed

by the diameter if^A^and the other part, JV»Z>i/; v \/ J

of the circumference, since otherwise, contrary to N. ^x M

the nature of the circle, a radius as AK would be ^"^ --^ I

less or ^eater than another radius as AH. \

*' (iii) Any radius MA can clearly be rectilineally produced only along a
single other radius AN^ since otherwise (contrary to the second Lemma) two
lines assumed straight, e.g. MAN^ MAH^ would have one and the same ^
common segment

" (iv) All diameters of the circle obviously cut one another in the centre
(Lemma 3 preceding), and they bisect one another there, by the go^eral
properties of the circle.

" From all this it is manifest that the diameter MAN divides its circle
and the circumference of it just exactly into two equal parts, and the same
ma^ be generally asserted for every diameter whatsoever of the same circle ;
which was to be proved."

Simson observes that the property is easily deduced from in. 31 and 24 ;
for it follows from iii. 31 that toe two parts of the circle are ^similar
segments" of a circle (segments containing equal angles, in. Def. 11), and
from III. 24 that they are equal to one another.

Definition 18.

'HfUJcvicXuiy Sc Itm ro irtpvf^iktvov vyri^a. ^ro re rSjs Sta/icrpov fcot r^ ,
dxoAafi)3ako/Mn^ vtt* aMj^ ir€pi/^€p€ia%, Kivrpov 8c rov iffiucvKkiov r^ at^ i I
icoi rov kvkXov IcrrtV. y

A semicircle is the figure contained by the diameter and the circumference cut
off by it. And the centre of the semicircle is the same as that of the circle.

The last words, '*And the centre of the semicircle is the same as that (
of the circle," are added from Proclus to the definition as it appears in the
Mss. Scarburgh remarks that a semicircle has no centre, properly speaking, |
and thinks that the words are not Euclid's, but only a note by Proclus. I am \
however inclined to think that they are genuine, if only because of the very *
futility of an observation added by Proclus. He explains, namely, that the '
semicircle is the only plane figure that has its centre on its perimeter (!), "so
that you may conclude that the centre has three positions, since it may be i
within the figure, as in the case of a circle, or on the perimeter, as with the I
semicircle, or outside, as with some conic Unes (the single-branch hyperbola '
presumably)" ! >

Proclus and Simplicius point out that, in the order adopted by Euclid for
these definitions of figures, the first figure taken is that bounded by one line
(the circle), then follows that bounded by two lines (the semicircle), then the
triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes

1

:v

1. Deff. 18-21] NOTES ON DEFINITIONS 17—21 187

different kinds of figures bounded by two lines (pp. 159, 14 — 160, 9). Thus
they may be formed

(i) by circumference and circumference, e.g. (a) those forming angles, as
a /un^ (to fArfvo€t&€^) and the figure included by two arcs with convexities
outward, and (d) the angle-less (dycJi^iOF), as the figure included between two
concentric circles (the coronal) \

(2) by circumference and straight line, e.g. the semicircle or segments of
circles (a^iScc is a name given to those less than a semicircle);

(3) by "mixed" line and "mixed" line, e.g. two ellipses cutting one
another;

(4) by " mixed " line and circumference, e.g. intersecting ellipse and
circle ;

(5) by " mixed " line and straight line, e.g. half an ellipse.

Following Def. 18 in the mss. is a definition of a segment of a circle which
was obviously interpolated from 111. Def. 6. Proclus, Martianus Capella and
Boethius do not give it in this place, and it is therefore properly omitted.

Definitions 19, 20, 21.

19. Sx^fuira tifOvypafAfia ioTi rk vro cii^cuSv ir^icxoficva, rpiVXcvpa /acv
ra viro rprnVf rcrpairXcvpa Sc ra {fwo rc<r<rapci>v, woKvvkwpa 8c ra vro nk^iovtav ij
Ttavapiav €vOtiwv ir€pi€XPfi€V€L

20. Toiv &k rpivkwpmv <ryyiimwv Urivktvpov fiky Tpiyuiv6y iari ro ra« rpcts
urac ixpv xXcvpac, uroa'K€kU Sk to rac fivo fi6yas Icras Ix^vk vXcvpav, (ffcaXiTvov Sk
TO Ttts Tpcis dvurov^ l)(ov wktvpas, -^

21. *ETi Sc TOIV TpiirXcvp<Dv <r)(iffMariav dpOaytivtov /acv rplywyop Icrri to c^ov
6fArjv ywvlavy dfiPkvyiaviov 8c to I^of d/ifiktZay ycDFtiav, 6$vyiivtoy 8c to Tas TpciS

19. Rectilineal figures are those which are contained by straight lines^
trilateral figures being those contained by three^ quadrilateral those contained by

four^ and multilateral those contained by more than four straight lines,

20. Of trilateral figures, an equilateral triangle fx that which has its three
sides equals an isosceles triangle that which has two of its sides alone equcU^ and
a scalene triangle thcU which has its three sides unequal.

2 1. Further^ of trilateral figures^ a right-angled triangle is that which has
a right angle^ an obtuse-angled triangle that which has an obtuse angle^ and an
acute-angled triangle that which has its three angles acute,

19.
The latter part of this definition, distinguishing three-sided^ four-sided and
many-sided figures, is probably due to Euclid himself, since the words
TpiirXcvpov, TCTpan-Xcvpov and xoXvVXcvpok do not appear in Plato or Aristotle
(only in one passage of the Mechanics and of the Problems respectively does
even rcTpaVXcupov, quadrilateral^ occur). By his use of TCTpaVXcvpov,
quadrilateral, Euclid seems practically to have put an end to any ambiguity
in the use by mathematicians of the word TCTpayuvov, literally "four-angled
(figure)," and to have got it restricted to the square. Cf. note on Def. 22.

20.

Isosceles {uroaictkiii, with equal legs) is used by Plato as well as Aristotle.
Scalene (<rKakr/y6% with the varient frKakrpnj^) is used by Aristotle of a triangle
with no two sides equal: cf. also Tim. Locr. 98 b. Plato, Euthyphro 12 d.

\

i88 BOOK I [t DiPP. so, sa

applies the term '' scalene " to an odd number in contrast to ^ isosceles " used
of an even number. Proclus (p. i68, 24) seems to connect it with 9taiff^ to
limp ; others make it akin to irmAi^ aroohedy aslani. Apollonius uses the
same word " scalene '* of an obUpti circular cone.

Triangles are classified, first with reference to their sides, and then with
reference to their angles. Produs points out that seven distinct species of
triangles emerge: (i) the equiiiUirai triangle, (a) three species of isasteies
triangles, the r^;ht-angled, the obtuse-angM and the acute-angled, (3) the
same three' varieties of scalene triangles.

Proclus gives an odd reason for the dual classification according to sides
and angles, namely that Euclid was mindful of the ftct that it is not every
triangle that is trilateral alsa He explains this statement by reference
(p. 165, 22) to a figure which some called barb4iki (d«ctotiSiyv) while
Zenodorus odled it koUauhangkd (MocXoyvJMoc). Proclus mentions it again
in his note on i. 22 (p. 328, 21 sqq[.) as one 61 the paradoxes of f;eoinetr]|r,
observing that it is seen in the figure of that proposition. This "triangle" is
merely a quadrilateral with a re-entrant angle ; and the idea that
it has only three angles is due to the n(m-recognition of the
fourth angle (which is greater than two right angles) as being an
angle at all. Since Proclus speaks of \h<t fomr-sided trian^ as
''one of the paradoxes in geometry,'' it is perhaps not sdfe to
assume that the misconception underlying tl^ expression existed
in the mind of Proclus alone ; but there does not seem to be any evidence
that Zenodorus called the figure in question a triangle (cf. Pappus, ed.
Hultsch, pp. 1 1 54, 1206). I

Definition 23. j

Tcav 8i rcrpaxXcvpuiv axflfdrtav rtrpaywvov fiiv loriv, i UrAwXoffi^w ri ion I
fcoi 6p0oywtoVf Ircpofii^fccs &c^ i 6p6oyw¥iov fih^^ oIk uFoirXcvpov hiy ^fLpo9 8c, i I
laowXwfiw /livf cWmc 6p6oyw¥iO¥ 8^ (&o/aJSoci8<s Sk ro r&s Airtyavrtov wkwvpai re irai i
ytiwia,^ uras dXAi^Acus ^XP^i ^ ^^ laowXtvpov Itrrw ovt€ 6pOoyiayioy ra 8i wapa
ravra rerpdrrXtvpa rpa7rc(ia KaXturdm, I

Of quadrilateral figures^ a square is that which is both equilateral and right- k
angled; an oblong that which is right-angled but not equilateral; a rhombus \
that which is equilateral but not right-an^d; and a rhomboid that which has ^
its opposite sides and angles equal to one another but is neither equilateral nor \
right-angUd. And let quadrilaterals other than these be called trapezia. .

rcrfMywrov was already a square with the Pythagoreans (cf. Aristotle,
Metaph, 986 a 26), and it is so most commonly in Aristotle ; but in De anima <
II. 3, 414 b 31 it seems to be a quadraateral, and in Metaph, 1054 b 2, (
'' equal and equiangular rcrpayoiva," it cannot be anything else but quadri- '
lateral if "equiangular" is to have any sense. Though, by introducing {
TtrpanrXitvpw for any quadrilateral, Euclid enabled ambiguity to be avoided,
there seem to be traces of the older vague use of rcrpayuiov in much later
writers. Thus Heron (Def. 104) speaks of a cube as "contained by six equi- I
lateral and equiangular r^rpaytava" and Proclus (p. 166, 10) adds to his !
remark about the " four-sided triangle ** that " you might have ra-paytava with
more than the four sides," where TtTpdytova can hardly mean squares. '

Ircpdfii^iccf, oblong (with sides of different length)^ is also a Pythagorean term.

The word right-angled (^oywiov) as here applied to quadrilaterals
must mean rectangular (Le., practically, having all its angles right angles);
for, although it is tempting to take the word in the same sense for a

i

i

I. Def. 22] NOTES ON DEFINITIONS 20—22 189

square as for a triangle (i.e. '* having one right angle "), this will not do in the
case of the oblong, which, unless it were stated that (Aree of its angles are
right angles, would not be sufficiently defined.

If it be objected, as it was by Todhunter for example, that the definition
of a square assumes more than is necessary, since it is sufficient that, being
equilateral, it should have one right angle, the answer is that, as in other cases,
the superfluity does not matter from Euclid's point of view ; on the contrary,
the more of the essential attributes of a thing that could be included in its
definition the better, provided that the existence of the thing defined and its
possession of all those attributes is proved before the definition is actually
used ; and Euclid does this in the case of the square by construction in i. 46,
making no use of the definition before that proposition.

The word rfunnbus (po/i^os) is apparently derived from ^cfijSoi^ to turn
roufid and round^ and meant among other things a spinning-top. Archimedes
uses the term solid rhombus to denote a solid figure made up of two right
cones with a common circular base and vertices turned in opposite directions.
We can of course easily imagine this solid generated by spinning \ and, if the
cones were equal, the section through the common axis would be a plane
rhombus, which would also be the apparent form of the spinning solid to the
eye. The difficulty in the way of supposing the plane figure to have been
named after the solid figure is that in Archimedes the cones forming the solid
are not necessarily equal. It is however possible that the solid to which the
name was originally given was made up of two equal cones, that the plane
rhombus then received its name from that solid, and that Archimedes, in
taking up the old name again, extended its signification (cf. J. H. T. Miiller,
Beitrdge zur Terminologie der griechischen Mathematiker^ i860, p. 20).
Proclus, while he speaks of a rhombus as being like a shaken, Le. deformed,
square, and of a rhomboid as an oblong that has been moved, tries to explain
the rhombus by reference to the appearance of a spinning square {r^pdyw^v

jk}flpOVfL€VOv).

It is true that the definition of a rhomboid says more than is necessary in
describing it as having its opposite sides and angles equal to one another.
The answer to the objection is the same as the answer to the similar objection
to the definition of a square.

Euclid makes no use in the Elements of the oblongs the rhombus^ the
rhomboid^ and the trapezium. The explanation of his inclusion of definitions
of the first three is no doubt that they were taken from earlier text-books.
From the words ^'let quadrilaterals other than these be called trapezia," we
may perhaps infer that this was a new name or a new application of an old
name.

As Euclid has not yet defined parallel lines and does not anywhere
define a parallelogram^ he is not in a position to make the more elaborate
classification of quadrilaterals attributed by Proclus to Posidonius and
appearing also in Heron's Definitions. It may be shown by the following
diagram, distinguishing seven species of quadnlaterals.

Quadrilaterals

parallelograms non-parallelograms

rectangular non-rectangular two sides parallel no sides parallel

I I (trapetium) (trapfioi4i)

I — ' — I I • 1 I ' 1

square pbhn^ rhombus rhotubcid isosceles trapetium ualeuo trapeuum

I

I90 BOOK I [i. Deff. 33, S3

It will be observed that^ while Euclid in the above definition classes as
trapezia all quadrilaterals other than iquares, obl(»igi, rfaombi, and rhomboid^
the word is in this classification restncted to quadrilaterals having two sides
(only) parallel, and trapezoid is used to denote the rest £uclid appears to
have used trapezium in the restricted sense of a quadrilateral with two sides
parallel in his book xc^m ScoipccrcMr (on divisions of figures). Archimedes
uses it in the same sense, but in one place describes it more precisely as a
trapezium with its two sides parallel.

Definition 23.

napoAXiyAoi clcru^ tiSuax^ oTrtvcf If ny uifT^ hr a r A ff oSooi fcat Ik^oXAi^^mmu
CIS air€tpov 1^* kicdrtpa ra yuifpii hn fiifiirMfio. uvf/ariwrowrw AAAijXaif.

Parallel straight tines are straight Ufus whieh^ being in the same plane emd
being produced indefinitely in bath Unctions^ do not meet one another in either
dirution,

IlapaAAi^Xof (alongside one another) written in one word does not appear
in Plato ; but with Aristotle it was already a familiar term.

€h aircifK>F cannot be translated ''to infinity" because these words might
seem to suggest a r^n or place infinitely distant, whereas ck avcipor, which
seems to be used indifferently with hr mipoy, is adverbial, meaning ''without
limit," i.e. *' indefinitely." Thus the expr^sion is used of a magnitude beiog
"infinitely divisible," or of a series of terms extending without limit

In both directions^ c^* indrtfia, ra §i4pi^ literally *' towards both the parts"
where *'parts" must be used in the sense of "regions" (cf Thuc ii. 96).

It is clear that with Aristotle the general notion of parallels was that of
straight lines which do not meet^ as in Euclid : thus Aristotle discusses the
question whether to think that parallels do meet should be called a
geometrical or an ungeometrical error {Anal, post, i. 12, 77 b 22), and (more
interesting still in relation to Euclid) he observes that there is. nothing
surprising in different hypotheses leading to the same error, as one might
conclude that parallels meet by starting from the assumption, either (a) that
the interior (angle) is greater than the exterior, or (b) that the angles of a
triangle make up more than two right angles (Anal, prior. 11. 17, 66 a 11).

Ajiother definition is attributed by Proclus to Posidonius, who said that
^^ parallel lines are tlwse which^ {being) in one plane^ neither converge nor diverge^
but have all the perpendiculars equal which are drawn from the points of one
line to the other ^ while such (straight lines) as make the perpendiculars less and
less continually do converge to one another ; for the perpendicular is enough
to define (opt^ciF hdvaraC) the heights of areas and the distances between lines.
For this reason, when the perpendiculars are equal, the distances between the
straight lines are equals but when they become greater and less, the interval is
lessened, and the straight lines converge to one another in the direction in
which the less perpendiculars are" (Proclus, p. 176, 6—17).

Posidonius' definition, with the explanation as to distances between straight {
lines, their convergence and divergence, amounts to the definition quoted by !
Simplidus (an-Nairizi, p. 25, ed. Curtze) which described straight lines as |
parallel if when they are produced indefinitely both ways^ the distance between
themy or the perpendicular drawn from either of them to the other ^ is always
equal and not different. To the objection that it should be prorved that the
distance between two parallel lines is the perpendicular to them Simplicius

•^1

1. Def. 23] NOTES ON DEFINITIONS 22, 23 191

replies that the definition will do equally well if all mention of the perpen-
dicuiar be omitted and it be merely stated that the distance remains equal,
although " for proving the matter in Question it is necessary to say that one
straight line is perpendicular to both'' (an-Nairizi, ed. Besthom-Heiberg, p. 9).
He then quotes the definition of "the philosopher Aganis": ^^Parailei
straight Ones are straight lines^ situated in the same plane^ the distance between
whichj if they are produced iiidefiniidy in both directions at the same time^ is
everywhere the sameJ' (This definition forms the basis of the attempt of
"Aganis" to prove the Postulate of Parallels.) On the definition Simplicius
remarks that the words ''situated in the same plane'' are perhaps unnecessary,
•since, if the distance between the lines is eve^where the same, and one does
not incline at all towards the other, they must for that reason be in the same
plane. He adds that the ''distance" referred to in the definition is the
shortest line which joins things disjoined.. Thus, between point and point,
the distance is the straight line joining them ; between a point and a straight
line or between a point and a plane it is the perpendicular drawn from the pomt
to the line or plane; "as regards the distance between two lines, that distance
b, if the lines are parallel, one and the same, equal to itself at all places on
the lines, it is the shortest distance and, at all places on the lines, perpendicular
to both " (i^k/. p. 10).

The same idea occurs in a quotation by Proclus (p. 177, 11) from
Geminus. As part of a classification of lines which do not meet he observes :
" Of lines which do not meet, some are in one plane with one another, others
not. Of those which meet and are in one plane, sofne are always the same
distance from one another^ others lessen the distance continually, as the hyper-
bola (approaches) the straight line, and the conchoid the straight line (i.e. the
asymptote in each case). For these, while the distance is being continually
lessened, are continually (in the position oQ not meeting, though they converge
to one another ; they never converge entirely, and this is the most paradoxical
theorem in geometry, since it shows that the convergence of some lines is non-
convergent. But of lines which are always an equal distance apart, those
which are straight and never make the (distance) between them smaller, and
which are in one plane, are parallel."

Thus the eguidistance-thieory of parallels (to which we shall return) is very
fully represented in antiquity. I seem also to see traces in Greek writers of a
conception equivalent to the vicious directian-Htieory which has been adopted
in so many modem text-books. Aristotle has an interesting, though obscure,
allusion in Ana/, pricr. 11. 16, 65 a 4 to a, petitio principii committed by "those
who think that diey draw parallels " (or " establish the theory of parallels,"
which is a possible translation of ra« ira^KiXAi^Xov« ypa^civ): "for they un-
consciously assume such things as it is not possible to demonstrate if parallels
do not exist" It is clear from this that there was a vicious circle in the then
current theory of parallels ; something which depended for its truth on the
properties of parallels was assumed in the actual proof of those properties,
e.g. that the three angles of a triangle make up two right angles. This is not
the case in Euclid, and the passage makes it clear that it was Euclid himself
who got rid of the petitio principii in earlier text-books by formulating and
premising before i. 39 the famous Postulate 5^ which must ever be r^arded
as among the most epoch-making achievements in the domain of geometry.
But one of the commentators on Aristotle, Philoponus, has a note on the
above passage purporting to give the specific character of the petitio principii
alluded to; and it is here that a direction-theory of parallels may be hinted at.

192 BOOK I [i. DBr. 13

whether Philoponus is or is not rig^t in supposing Aat this was what Aristotle
had in mind. Philoponus says: ''The same thinff is done by those iriio dnw
parallels, namely begging the original question; for thc^ will have it tfiat it is
possible to draw parallel straight lines from the mendian circle, and they
assume a point, so to say, falluiff on the plane of that circle and thus they
draw the straight lines. And what was sought is thereby assumed; for he
who does not admit the genesis of the parallels will not admit the point
referred to either." What is meant is, I think, somewhat as follows. Given
a straight line and a point through which a parallel to it is to be drawn, we
are to suppose the given straight line placed in the plane of the meridian. J
Then we are told to draw through the given point anodier straight line in the ^
plane of the meridian (strictly speaking it should be drawn in a plane pualld k
to the plane of the meridian, but the idea is that, compared with the sixe of |
the meridian circle, the distance between the point and the straight line is {
negligible) ; and this, as I read Philoponus, is supposed to be equivalent to j
assuming a very distant point in the meridian plane and joining the given J
point to it But obviously no ruler would stretch to such a pomt, and the
objector would say that we cannot really direct a straight line to die assumed
distant point except by drawing it, widiout more ado, paralld to the given
straight line. And herein is the pttiHo frmcifii, I am confirmed in seeing
in Philoponus an allusion to a dineiiOH'thearj by a remark of Schotten on a
similar reference to the meridian plane supposed to be used by advocates of .
that theory. Schotten is aiguing that direction is not in itself a conception '
such that you can predicate Me direction of two different lines. ** If any one
should reply that nevertheless many lines can be conceived which aU have the
direction from north to south^^ he replies that this represents only a nominal,
not a real, identity of direction.

Coming now to modem times, we may classify under three groups
practically all the different definitions that have been given of parallels
(Schotten, op, cit. 11. p. 188 sqq.).

(i) Parallel straight lines hm*e no point common^ under which general
conception the following varieties of statement may be included :

{a) they do not cut one another^ ^

{b) they meet at infinity^ or ^

(c) they have a common point at infinity. \

(2) Parallel straight lines have the same^ or lihe^ direction or directions^
under which class of definitions must be included all those which introduce '
transversals and say that the parallels mahe equal angles with a transverscU. .

(3) Parallel straight lines have the distance t^etween thetn constant \ \
with which group we may connect the attempt to explain a parallel as the >
geometrical locus of all points which are equidistant from a straigkt line. {

But the three points of view have a good deal in common ; some of them
lead easily to the others. Thus the idea of the lines having no point common
led to the notion of their having a common point at infinity, through the |
influence of modem geometry seeking to embrace different cases under one f
conception ; and then again the idea of the lines having a common point at ^
infinity might suggest their having the same direction. The " non-secant " f
idea would also naturally lead to that of equidistance (3), since our
observation shows that it is things which come nearer to one another that
tend to meet, and hence, if lines are not to meet, the obvious thing is to see
that they shall not come nearer, i.e. shall remain the same distance apart.

\

I

I. D«F. 23] NOTE ON DEFINITION 23 I93

We will now take the three groups in order.

(i) The first observation of Schotten is that the varieties of this group
which regard parallels as (a) meeting at infinity or (d) having a common
point at mfinity (first mentioned apparentlv by Kepler, 1604, as a '*fa^on de
parler" and then used by Desaigues, 1639) are at least unsuitable definitions
tor elementary text-books. How do we know that the lines cut or meet at
infinity? We are not entitled to assume either that they do or that they do
not, because "infinity'' is outside our field of observation and we cannot verify
either. As Gauss says (letter to Schumacher), " Finite man cannot claim to
be able to regard the infinite as something to be grasped by means of ordinary
methods of observation." Steiner, in speaking of the rays passing through a
point and successive points of a straight line, observes that as the point of
intersection gets further away the ray moves continually in one and the same
direction ("nach einer und derselben Richtung hin"); only in one position,
that in which it is parallel to the straight line, "there is no real cutting^*
between the ray and the straight line ; what we have to say is that the ray is
*^ directed towards the infinitely distant paint an the straight line.^^ It is true
that higher geometry has to assume that the lines do meet at infinity: whether
such lines exist in nature or not does not matter (just as we deal with "straight
lines " although there is no such thing as a straight Une). But if two lines do
not cut at any finite distance, may not the same thing be true at infinity also ?
Are lines conceivable which would not cut even at infinity but always remain
at the same distance from one another even there ? Take the case of a line
of railway. Must the two rails meet at infinity so that a train could not stand
on them there (whether we could see it or not makes no difference)? It
seems best therefore to leave to higher geometry the conception of infinitely
distant points on a line and of two straight lines meeting at infinity, like
imaginary points of intersection, and, for the purposes of elementaiy geometry,
to rely on the plain distinction between "parallel" and "cutting" which
average human mtelligence can readily grasp. This is the method adopted
by Euclid in his definition, which of course belongs to the group (i) of
definitions regarding parallels as non-secant

It is significant, I think, that such authorities as Ingrami {Elementi di
geometria^ 1904) and Enriques and Amaldi (Elementi di geometria^ 1905)1
after all the discussion of principles that has taken place of late years, give
definitions of parallels equivalent to Euclid's : " those straight lines in a plane
which have not any point in common are called parallels." Hilbert adopts
the same point of view. Veronese, it is true, takes a different line. In his
great work Fondamenti di geametria, 1 891, he had taken a ray to be parallel to
another when a point at infinity on the second is situated on the first ; but he
appears to have come to the conclusion that this definition was unsuitable for
his Elementi. He avoids however giving the Euclidean definition of parallels
as "straight lines in a plane which, though produced indefinitely, never meet,"
because "no one has ever seen two straight lines of this sort," and because
the postulate generally used in connexion vnth this definition is not evident in
the way that, m the field of our experience, it is evident that only one straight
line can pass through two points. Hence he gives a different definition, for
which he claims the advantage that it is independent of the plane. It is
based on a definition of figures " opposite to one another with respect to a
point" (or reflex figures). "Two figures are opposite to one another with
respect to a point C7, e.g. the figures ABC ... and A'ffC.,.^ M to every point
of the one there corresponds one sole point of the other, and if the segments

H. s. 13

t94 6001C t [i. DBF. tj

OA^ OB^ OCy ... joining the points of one figure to C7 are respectively equal
and opposite to the segments 0A\ OB^ 0C\ ... joining to O the corresponding
points of the second " : then, a iramsverstU of two straight lines being any
segment having as its extremities one point of one line and one point of the
other, " two straight lines are aUlei parallel if one of them contains two foinis
opposite to two points of the other with reject to the middle point of a common
transverscU,^^ It is true, as Veronese says, that the paiallek so defined and the
parallels of Euclid are in substance the same; but it can hardly be said diat
the definition gives as good an idea of the essential nature of parallels as does
Euclid's. Veronese has Xjoprove^ of course, that his parallels have no point in
common, and his ''Postulate of Parallds" can hardly be called more evident
than Euclid's : " If two straight lines are parallel, they are figures opposite to
one another with respect to the middle points of all their transversal segments."

(2) The direction-xheoiy.

The fallacy of this theory has nowhere been more completely exposed
than by C. L. Dodgson (EuJid and his modem Moals, 1879). According to
Killing {Einfuhrung in die Grundlagm der Geometrie^ i. p. 5) it would appear
to have originated with no less a person than Leibniz. In the text-books
which employ this method the notion of direetion appears to be regarckd as a
primary, not a derivative notion, since no definition is given. But we ought
at least to know how the same direction or like directions can be recognised
when two different straight lines are in question. But no answer to this
question is forthcoming. The fact is that the whole idea as applied to non-
coincident straight lines is derived firom knowledge of the properties of
parallels ; it is a case of explaining a thing by itself. The idea of parallels
being in the same direction perhaps arose from the conception of an angle as
a differefue of direction (the hoUowness of which has already been expmed) ;
sameness of direction for parallels follows from the same "difference of
direction" which both exhibit relatively to a third line. But this is not
enough. As Gauss said {Werhe, iv. p. 365), " If it [identity of direction] is
recognised by the equality of the angles formed with one third straight bne,
we do not yet know vnthout an antecedent proof whether this same equality
will also be found in the angles formed with a fourth straight line " (and any
number of other transversals) ; and in order to make this theory of parallels
valid, so far from getting rid of axioms such as Euclid's, you would have to
assume as an axiom what is much less axiomatic, namely that "straight lines
which make equal corresponding angles vnth a certain transversal do so with
any transversal" (Dodgson, p. loi).

(3) In modem times the conception of parallels as equidistant straight
lines was practically adopted by Clavius (the editor of Euclid, bom at
Bamberg, 1537) and (according to Saccheri) by Borelli {Euclides restitutus^
1658) although they do not seem to have defined parallels in this way.
Saccheri points out that, before such a definition can be used, it has to
be proved that " the geometrical locus of points equidistant fi^m a straight J
line is a straight line." To do him justice, Clavius saw this and tried to
prove it: he makes out that the locus is a straight line according to the
definition of Euclid, because "it lies evenly with respect to all tiie points
on it"; but there is a confusion here, because such "evenness" as the locus
has is with respect to the straight line fi^m which its points are equidistant,
and there is noUiing to show that it possesses this property with respect
to itself. In fact the theorem cannot be proved vrithout a postulate.

J

I I. Post, i] N0T£ ON t>OStULATfi i tpj

Postulate i.

HinioBia dwo toitos arjfjLtlov hci xav (rqfUiov cMciav ypafjLfirjy ayayctv.
Let the following be postulated: to draw a straight line from any point to
any point.

From any point to any point. In general statements of this kind
the Greeks did not say, as we do, "a/iy point," ^*any triangle" etc., but
^* every point," ^^ every triangle" and the like. Thus the words are here
literally "from every point to every point." Similarly the first words of
Postulate 3 are " with every centre and distance," and the enunciation, e.g., of
I. 1 8 is " In every triangle the greater side subtends the greater angle."

It will be remembered that, according to Aristotle, the geometer must in
general assume what a thing is, or its definition, but must prove that it is,
i.e. the existence of the thing corresponding to the definition : only in the case
of the two most primary things, points and lines, does he assume, without
proof, both the definition and the existence of the thing defined. Euclid has
mdeed no separate assumption affirming the existence oi points such as we find
nowadays in text-books like those of Veronese, Ingrami, Enriques, "there exist
distinct points" or "there exist an infinite number of points." But, as re-
gards the only lines dealt with in the Elements^ straight lines and circles,
existence is asserted in Postulates i and 3 respectively. Postulate i however
does much more than (i) postulate the existence of straight lines. It is
(2) an answer to a possible objector who should say that you cannot, with the
imperfect instruments at your disposal, draw a mathematical straight line at all,
and consequently (in the words of Aristotle, Anal, post. i. 10, 76 b 41) that
the geometer uses ^se hypotheses, since he calls a line a foot long when it is
not or straight when it is not straight. It would seem (if Gherard's translation
is right) that an-Nairizi saw that one purpose of the Postulate was to refute
this criticism : " the utility of the first three postulates is (to ensure) that the
weakness of our equipment shall not prevent (scientific) demonstration"
(ed. Curtze, p. 30). The fact is, as Aristotle says, that the geometer's demon-
stration is not concerned with the particular imperfect straight line which he
has drawn, but with the ideal straight line of which it is the imperfect
represer^tation. Simplicius too indicates that the object of the Postulate is
rather to enable the drawing of a mathematical straight line to be imagined
than to assert that it can actually be realised in practice : " he would be a
rash person who, taking things as they actually are, should postulate the
drawing of a straight line from Aries to Libra."

There is still something more that must be inferred from the Postulate
combined with the definition of a straight line, namely (3) that the straight
line joining two points is unique : in other words that, tf tivo straight lines
("rectilineal segments," as Veronese would call them) luwe the same extremities^ -
they must coinade throughout their length. The omission of Euclid to state
this in so many words, though he assumes it in i. 4, is no doubt answerable for
the interpolation in the text of the equivalent assumption that two straight
lines cannot enclose a space^ which has constantly appeared in mss. and editions
of Euclid, either among Axioms or Postulates. That Postulate i included it,
by conscious implication, is even clear from Proclus* words in his note on l 4
(p. 239, 16) : "therefore two straight lines do not enclose a space, and it was
with knowledge of this fact that the writer of the Elements said in the first of
his Postulates, to draw a straight line from any point to any pointy implying
that it is ^M^ straight line which would always join the two points, not two.^

13— »

I

19^

BOOK I

[l Post.

i»s

Proclus attempts in the same note (p. 339) to P^yoe that two straight lines
cannot enclose a space, using as his bws the definition of the diameter of a
circle and the theorem, stat^ in it, that any diameter divides tfie cirde into
two equal parts. '

Suppose, he says, ACB^ ADB to be two straight lines enclosing a q»oe.
Produce them (beyond B) indefinitely. With centre B
and distance AB describe a circle, cutting the lines so
produced va F^ E respectively.

Then, since ACBF, ADBE are both diameters
cutting off semi-circles, the arcs AE^ AEF are equal :
which is impossible. Therefore etc.

It ¥nll be observed, however, that the straight lines
produced are assumed to meet the circle given in two
different points E^ F^ whereas, for anything we know,
Ey F might coincide and the straight lines have three common points. The
proof is therefore delusive.

Saccheri gives a different {xoof. From Euclid's definition of a straight
line as that which lies evenly with its points he infers that, when
such a line is turned about its two extremities, which remain fixed,
all the points on it must remain throughout in the same position, and
cannot take up different positions as the revolution proceeds. " In
this view of the straight line the truth of the assertion that two
straight lines do not enclose a space is obviously involved. In fact,
if two lines are given which enclose a s^ce, and of which the two
points A and X are the common extremities, it is easily shown that
neither, or else only one, of the two lines is straight"

It is however better to assume as a postuiate the fact, inseparably
connected with the idea of a straight line, that there exists onfy one straight
line containing two given points^ or, if two straight lines have two points in
commotiy they coiticide throughout.

Postulate 2.

Kal icaetpaxT\kivyfv tMtiav Kara to (twc^^ i'f* cMctac ^ic/SaXciy.

To produce a finite straight line continuously in a straight line,

I translate icvtrmaxr\tivrjpf by finite^ because that is the received equivalent,
and because any alternative word such as limited^ terminated^ if applied to a
straight line, would equally fail to express what modem Italian geon^eters aptly
call a rectilineal segment^ that is, a straip;ht line having two extremities.

Just as Post. I asserting the possibility of drawing a straight line from any
one point to another must be held to declare at the same time that the
straight line so drawn is unique, so Post 2 maintaining the possibility of
producing a finite straight line (a "rectilineal segment ") continuously m a
straight hne must also be held to assert that the straight line can only be
produced in one way at either end, or that the produced part in either
direction is unique ; in other words, that two straight lines cannot have a
common segment. This latter assumption is not expressly appealed to by
Euclid until xi. i. But it is needed at the very beginning of Book'i. Proclus
(p. 314, 18) says that 2^no of Sidon, an Epicurean, maintained that the very
first proposition i. i requires it to be admitted that *' two straight lines cannot
have the same segments " ; otherwise AQ BC might meet before they arrive
at C and have the rest of their length common, in which case the actual
triangle formed by them and AB would not be equilateral. The assumption
that two straight lines cannot have a common segment is certainly necessary
in I. 4, where one side of one triangle is placed on that side of the other

(

j L Post. 2]

NOTES ON POSTULATES i. 2

197

triangle which is equal to it, and it is inferred that the two coincide throughout

their length : this would by no means follow if two straight lines could have a

common segment Proclus (p. 215, 24), while observing that Post. 2 clearly

indicates that the produced portion must be one^ attempts to prove it, but

unsuccessfully. Both he and Simplicius practically

use the same argument. Suppose, says Proclus,

that the straight lines ACy AD have AB as a

common segment. With centre B and radius BA

describe a circle (Post 3) meeting AC^ AD in

C, D. Then, since ABC is a straight line through

the centre, AEC is a semi-circle. Similarly, ABD

being a straight line through the centre, AED is a

semi-circle. Therefore AEC is equal to AED\

which is impossible.

Proclus observes that 2^no would object to this proof as really depending
on the assumption that "two circumferences (of circles) cannot have one
portion common " ; for this, he would say, is assumed in the common proof
by superposition of the fact that a circle is bisected by a diameter, since that
proof takes it for granted that, if one part of the circumference cut off by the
diameter, when applied to the other, does not coincide with it, it must neces-
sarily fall either entirely outside or entirely inside it, whereas there is nothing
to prevent their coinciding, not altogether, but in part only ; and, until you
really prove the bisection of a circle by its diameter, the above proof is not
valid. ' Posidonius is represented as having derided Zeno for not seeing that
the proof of the bisection of a circle by its diameter goes on just as well if the
circumferences fail to coincide in part only. But the true objection to the
proof above given is that the proof of the bisection of a circle by any diameter
i/f^'^ assumes that two straight lines cannot have a common segment; for, if
we wish to draw the diameter of a circle which has its extremity at a given point
of the circumference we have to join the latter point to the centre (Post i) and
then to produce the straight line so drawn till it meets the circle again (Post 2),
and it is necessary for the proof that the produced part shall be unique.

Saccheri adopted the proper order when he gave, first the proposition that
two straight lines cannot have a common s^ment, and after that the
proposition that any diameter of a circle bisects the circle and its circumference.

Saccheri's proof of the former is very interesting as showing; the thorough-
ness of his method, if not at the end entirely convincing. It is in five stages
which I shall indicate shortly, giving the full argument of the first only.

Suppose, if possible, that AX is a common segment of both the straight
lines AXB^ AXC^ in one plane, produced beyond
X. Then describe about X as centre, with radius
XB or XC^ the arc BMC^ and draw through X to
any point on it the straight line XM.

(1) I maintain that, with the assumption
made, the line AXM is also a straight line which
is drawn from the point A to the point X and pro-
duced beyond X.

For, if this line were not straight, we could draw
another straight line AM which for its part would
be straight This straight line will either (a) cut one
of the two straight lines XB^ XC in a certain point
K or (b) enclose one of them, for instance XBy in
the area bounded by AX^ XMsind APLM.

193 BOOK I [lPost. i

But the first alternative (a) obviously contndicU the forq^oiiig lemma rtliat
two straight lines cannot enclose a space^ since in that case the two lines
AXJ^, ATK^ which by hypothesis are straight, would enclose a space.

The second possibility ijf) is at once seen to invdve a similar abaurdity.
For the straight line XB must, when produced beyond B^ ultimatel]r meet
APLM in a point Z. Consequently the two lines AXBL^ APL^ which by
hypothesis are straight, would again enclose a space. If however we were to
assume that the straight line XB produced beyond B will ultimately meet
either the straight line XMor the straight line XA in another point, we should
in the same way arrive at a contradiction.

From this it obviously follows that, on the assumption made, the line
AXM\% itself the straight line which was drawn from the point ^ to the point
M'y and that is what was maintained.

The remaining stages are in substance these.

(ii) Iftke straight line AXB, regarded as rigid^ revohes about AX as axis^
U cannot assume two more positions in the same fiane^ so tkat^ for exam^e^ in
one position XB should coincide with XC, and in the other with XM. ^

[This is proved by considerations of symmetry. AXB cannot be altogether
** similar or equal to " AXC^ if viewed from the same side (left or ri^t) of
both : otherwise they would coincide, which by hypothesis they do not But
there is nothing to prevent AXB viewed from one side (sav the left) being
'* similar or equal to '' AXC viewed from the other side (i.e. the right), so that
AXB can^ without any chanffe, be brought into the position AXC.

AXB cannot however tale the position of the other straight line AXAitm i
well If they were like on one side, they would coincide; if they were like on 1
opposite sides, AXM^ AXC would be like on the same side and therefore j
comdde.] I

(iii) The other positions of AXB during the revolution must be above or 4
below the original plane. 4

(iv) It is next maintained that there is a point l^ on the arc BC such that^ if 1
XD is drffwn^ AXD is not only a straight line but is such that viewed from the left I
side it is exactly ^^ similar or equaV^ to what it is when viavedfrom the right side, >

{Firsts it is proved that points M^ Fcan be found on the arc, corresponding \
in the same way as B^ C do, but nearer together, and of course AXM^ AXF
are both straight lines.

Secondly^ similar corresponding points can be found still nearer together,
and so on continually, until either (a) we come to one point D such that AXD \
is exactly iihe itself when the right and left sides are compared^ or (b) there are ;
two ultimate points of this sort M^ F^ so that both AXM^ AXF have this
property.

Thirdly^ (b) is ruled out by reference to the definition of a straight line. '

Hence (a) only is true, and there is only one point D such as described.]

(v) Lastly, Saccheri concludes that the straight line AXD so determined i
*' is alone a straight line, and the immediate prolongation from A beyond X to
/>," relying again on the definition of a straight line as 'Mying evenly."

Simson deduced the proposition that two straight lines cannot have a
common segment as a corollary from i. 1 1 ; but his argument is a complete
petitio principii^ as shown by Todhunter in his note on lliat proposition.

Produs (p. SI 7, lo) records an ancient proof also based on the proposition
I. 1 1. Zeno, he says, propounded this proof and then critidsed it

I. Post- 2, 3] NOTES ON POSTULATES 2, 3 199

Suppose that two straight lines AC^ AD have a common segment AB^ and
let BE be drawn at right angles to AC.

Then the angle BBC is right.

If then the angle EBD is also right, the two
angles will be equal : which is impossible.

If the angle EBD is not right, draw ^^at right
angles to AD \ therefore the angle FBA is right.

But the angle EBA is right

Therefore the angles EBA^ FBA are equal :
which is impossible.

Zeno objected to this, says Proclus, because it assumed the later pro-
position I. 1 1 for its proof. Posidonius said that there was no trace of such
a proof to be found in the text-books of Elements, and that it was only invented
by Zeno for the purpose of slandering contemporary geometers. Posidonius
maintains further that even this proof has something to be said for it. There
must be some straight line at right angles to each of the two straight lines A C,
AD (the very definition of right angles assumes this): ^^ suppose /Aen U happens
to be the straight line we have set upJ* Here then we have an ancient instance
of a defence of hypothetical construction^ but in such apologetic terms (" it is
possible to say something even for this proof") that we may conclude that in
general it would not have been accepted by geometers of that time as a
Intimate means of proving a proposition.

Todhunter proposed to deduce that two straight lines cannot have a
common segment from i. 13. But this will not serve either, since, as before
mentioned, the assumption is really required for i. 4.

It is best to make it a postulate.

Postulate 3.

Kal Tovrl iccFrpcp jcal Zuurrrnijari kvkXjO¥ ypa^cotfai.

To describe a circle with any centre and distance.

In this case Euclid's Jext has the passive of the verb: '*a circle can be
drawn " ; Proclus however has the active (ypJ^oi) as Euclid has in the first
two Postulates.

Distance^ iuKm/fiaTL This word, meaning ** distance *' quite generally (cf.
Arist Metaph. 1055 a 9 *'it is between extremities that distance is greatest,"
ibid. 1056 a 36 '' things which have something between them, that is, a certain
distance "), and also " distance " in the sense of '' dimension " (as in " space
has three dimensions, length, breadth and depth," Arist. Physics iv. i, 209 a 4),
was the regular word used for describing a circle with a certain radius^ the
idea being that each point of the circumference was at that distance from the
centre (cf. Arist. Meteorologica in. 5, 376 b 8 : "if a circle be drawn. ..with
distance Mil "). The Greeks had no word corresponding to radius : if they
had to express it, they said "(straight lines) drawn from the centre" (al Ik rov
Kivrpov, Eucl. III. Def. I and Prop. 26; Meteorologica 11. 5, 362 b i has the full
phrase ol Ik tov mtyrpov dyofuyiu ypofifjuoi),

Mr Frankland observes that it would be remarkable if, unlike Postulates i
and 2, this Postulate implied merely what it says, that a circle can be drawn
with any centre and distance. We may r^ard it, if we please, as helping to the
complete delineation of the Space which Euclid's geometry is to investigate
formally. The Postulate has the effect of removing any restriction upon the
size of the circle. It may (i) be indefinitely small, and this implies that space
is continuous, not discrete, with an irreducible minimum distance between

soo BOOK I [l Post 3, 4

contiguous pomts in it (a) The drde may be indefinitely bu8e» wfaidi
implies the fundamental hypothecs of h^miudi of space. This last assumed
characteristic of space is essential to the proof of i. i6^ a theorem not
universally valid in a space whidi is unbounded in extent but finite in sise. It
would however be unsafe to suppose that Euclid foresaw the use to which his
Postulate might thus be put, or formulated it with such an intention.

Postulate 4.

Koi irauiif ths 6p$a9 yta^tom Itrat oXXipXaif Aoi.

ITiat all right angles are equal to 4me another.

While this Postulate asserts the essential truth that a risht angle is a
determinate magnitude so that it really serves as an invariable standard by
which other (acute and obtuse) angles may be measured, much more than
this is implied, as will easily be seen firom the following consideratioiL ^ If the
statement is to be proved^ it can only be proved b^ the method of applying one
pair of right angles to another and so arguing their equality. But this method
would not be valid unless on the assumption of the invaruMlity ofjigiirest
which would therefore have to be asserted as an antecedent postulate. Euclid
preferred to assert as a postulate, directly, the fact that all right angles are
equal; and hence his postulate must be tidten as equivalent to me principle of
invariability of figures or its equivalent, the homogeneity of space.

According to Proclus, Geminus held that this Postulate should not be
classed as a postulate but as an axiom, since it does not, like the first three
Postulates, assert the possibility of some amstntction but expresses an essential
property of right angles. Produs further observes (p. 188, 8) that it is not a
postulate in Aristotle's sense either. (In this I think he is wrong, as exjdained
above.) Proclus himself, while regarding the assumption as axiomatic (** the
equality of right angles suggests itself even by virtue of our common notions"),
is prepared with a proof, if such is asked for.

Let ABC, DEF be two right
angles.

If they are not equal, one of them
must be the greater, say ABC.

Then, if we apply DE to AB, EF H-
will fall within ABC, as BG.

Produce CB to H. Then, since
ABC is a right angle, so is ABH, and the two angles are equal (a right angle
being by definition equal to its adjacent angle).

Therefore the angle ABH\% greater tlum the angle ABG.

Producing GB to K, we have similarly the two angles ABK, ABG both
right and equal to one another; whence the angle ABHS& less than the angle
ABG.

But it is also greater : which is impossible.

Therefore etc.

A defect in this proof is the assumption that CB, GB can each be
produced only in one way, and that ^9^ falls outside the angle ABH.

Saccheri's proof is more careful in that he premises a third lemma in
addition to those asserting (i) that two straight lines
cannot enclose a space and (2) that two straight lines
cannot have a common segment The third lemma is :
If two straight Unes AB, CXD meet one another at an
intermediate point X, they do not touch at that point, but
cut one another.

1

i

I

1. Post. 4]

NOTES ON POSTULATES 3, 4

aoi

M

Suppose now that DA standing on BAC makes the two angles DAB^
Z>^C equal, so that each is a right angle by the definition.

Similarly, let LHioxm with the straight line FHM the right angles LHFy
LHM.

Let DA^' HL be equal ; and sup-
pose the whole of the second figure
so laid upon the first that the point
^falls on A^ and L on Z>.

Then the straight line FHMmW
(by the third lemma) not touch the
straight line BC at A ; it will either

(a) coincide exactly with BC, or

(d) cut it so that one of its extremities, as F, will fall above [BC] and the
other, M^ below it.

If the alternative (a) is true, we have already proved the exact equality of
all rectilineal right angles.

Under alternative (b) we prove that the angle LHF, being equal to the
angle DAF, is less than the angle DAB or DAC, and a fortiori less than the
angle DAM ox LHM\ which is contrary to the hypothesis.

[Hence (a) is the only possible alternative, so that all right angles are
equal.]

Saccheri adds that it makes no difference if the angle DAF diverges
infinitely little from the angle DAB, This would equally lead to a conclusion
contradicting the hypothesis.

It will be observed that Saccheri speaks of ''the exact equality of all
rectilineal right angles." He may have had in mind the remark of Pappus,
quoted by Proclus (p. 189, 11), that the converse of
this postulate, namely that an angle which is equal
to a right angle is also right, is not necessarily true,
unless the former angle is rectilinecU. Suppose two
equal straight lines BA^ BC9X right angles to one
another, and semi-circles described on BA, BC
respectively as AEB, BDC in the figure. Then,
since the semi-circles are equal, tiiey coincide if
applied to one another. Hence the ''angles **
EBA, DBC are equal Add to each the " angle "
ABD ; and it follows that the lunular angle EBD is equal to the right angle
ABC. (Similarly, if BA, BC be inclined at an acute or obtuse angle, instead
of at a rfght angle, we find a lunular angle equal to an acute or obtuse angle.)
This is one of the curiosities which Greek commentators delighted in.

Veronese, Ingrami, and Enriques and Amaldi deduce the fact that all
right angles are equal from the equivalent fact that all fiat angles are equals
which is either itself assumed as a postulate or immediately deduced from some
other postulate.

HUbert takes quite a different line. He considers that Euclid did wrong
in placing Post 4 among "axioms." He himself, after his Group in. of
Axioms containing six relating to congruence, proves several theorems about
the congruence of triangles and angles, and then deduces our Postulate.

As to the raison iPitre and the place of Post 4 one thing is quite certain.
It was essential from Euclid's point of view that it should come before Post. 5,
since the condition in the latter that a certain pair of angles are together less
than two right angles would be useless unless it were first made clear Ihat
right angles are angles of determinate and invariable magnitude.

ao3 BOOK I [i. Post. 5 1

Postulate 5.

Kcu iay etc Svo c^ctat cMcia i§iwiwr€vaa rkt hrri^ mu hn rk aira fU/ni vtMit
Suo 6ft6mv iXaaaoya^ iroc^ iKPaXXofiiva% r&« Svo cMctat Jv* avcipor ov/cviVTCcr,

7>(a/, (/*« straight line falling on two straighl lines make ike interior angles
on the same side less than two right angles^ the two straight Knes^ if prodneed
indefinitely^ meet on that side on which an the angles less than the two rfght
angles.

Although Aristotle gives a dear idea of what he understood by a postulaie^
he does not give any instances from g^metry; still less has he an^ allusion
recalling the particular postulates found in Euclid. We naturally infer that
the formulation of these postulates was Euclid's own work. There is a more
positive indication of the originality of Postulate 5* since in the passage {Anal,
prior. II. 16, 65 a 4) quoted above in the note on the definition of parallels he
alludes to some petitio primipii involved in the theory of parallels current in
his time. This reproach was removed by Euclid when he laid down this
epoch-making Postulate. When we ocmsider the countless successive attempt
made through more than twenty centuries to prove the Postulate, many of
them by geometers of abili^, we cannot but admire the genius of the man
who conduded that such a hypothesisi which 'he found necessary to the
validity of his whole system of geometiy, wasjc^ly indem(mstmble.

From the very b^mning» as we know irom ProcIus| the Tostulate was
attacked as such, and attempts were made to prove it as a theorem or to get
rid of it by adopting some odier definition of jparallds; while in modem times
the literature of the subject is enormous. Riccardi {^aggju^ di una bibU^^fta
Euclidea^ Part iv., Bologna, 1890) has twenty quarto pages of titles of mono-
graphs relating to Post 5 between the dates 1607 and 1887. Max Simon
\Ueber die Entwichlung der Elementar-geometrie im XIX. Jahrhundtrt^ 1906)
notes that he has seen three new attempts, as late as 1891 (a century after
Gauss laid the foundation of non-Euclidean geometry), to prove the theory of
' parallels independently of the Postulate. Max Simon himself (pp. 53 — 61)
gives a large number of references to books or articles on the subject and
refers to the copious information, as to contents as wdl as names, con-
tained in Schotten's Inhalt und Methodt des planimetrischen Unterrichts^ IL

PP- 1*3— 33»-

This note will include some account of or allusion to a few of the most
noteworthy attempts to prove the Postulate. Only those of andent times, as
being less generally accessible, will be described at any length; shorter
references must suffice in the case of the modem geometers who have made
the most important contributions to the discussion of the Postulate and have
thereby, in particular, contributed most towards the foundation of the non-
Euclidean geometries, and here I shall make use prindpally of the valuable
Article 6, Sulla teoria delle parallele e sulk geomeirie nem-euclidee (by Roberto
Bonola), in Ouestioni riguardanti la geametria elementare (pp. 143 — 222).

Proclus (p. 191, 21 sqq.) states very clearly the nature of the first
objections taken to the Postulate.

*'This ought even to be struck out of the Postulates altogether; for it is a
theorem involving many difficulties, which Ptolemy, in a certain book, set
himsdf to solve, and it requires for the demonstration of it a number
of definitions as well as the(»ems. And the converse of it is actually
pioved by Eudid himsdf as a theorem. It may be that some would be

I

I. Post. 5] NOTE ON POSTULATE 5 203

1 deceived and would think it proper to place even the assumption in question
among the postulates as affording, in the lessening of the two right angles,
ground for an instantaneous belief that the straight lines converge and meet.
To such as these Geminus correctly replied that we have learned from the
very pioneers of this science not to have any regard to mere plausible imagin-
ings when it is a question of the reasonings to be included in our geometrical
doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of
a rhetorician as to accept mere plausibilities from a geometer; and Simmias is
made by Plato to say that he recognises as quacks those who fashion for
themselves proofs from probabilities. So in this case the fact that, when the
right angles are lessened, the straight lines converge is true and necessary;
but the statement that, since they converge more and more as they are pro-
duced, they will sometime meet is plausible but not necessary, in the absence
of some argument showing that this is true in the case of straight lines. For
the fact that some lines exist which approach indefinitely, but yet remain
non-secant (dav/LnrrcDroi), although it seems improbable and paradoxical, is
nevertheless true and fully ascertained with regard to other species of lines.
May not then the same thing be possible in the case of straight lines which
happens in the case of the lines referred to ? Indeed, until the statement in
the Postulate is clinched by proof, the facts shown in the case of other lines
may direct our imagination the opposite way. And, though the controversial
arguments against the meeting of the straight lines should contain much that
is surprising, is there not all the more reason why we should expel from our
body of doctrine this merely plausible and unreasoned (hypothesis) ?

"It is then clear from this that we must seek a proof of the present
theorem, and that it is alien to the special character of postulates. But how
it should be proved, and by what sort of arguments the objections taken to
it should be removed, we must explain at the point where the writer of the
Elements is actually about to recall it and use it as obvious. It vnll be
necessary at that stage to show that its obvious character does not appear
independendy of proof, -but is turned by proof into matter of knowledge."

Before passing to the attempts of Ptolemy and Proclus to prove the
Postulate, I should note here that Simplicius says (in an-NairIzi, ed. Besthom-
Heiberg, p. 119, ed. Curtze, p. 65) that this Postulate is by no means manifest,
but requires proof, and accordingly '* Abthiniathus " and Diodorus had
already proved it by means of many different propositions, while Ptolemy also
had explained and proved it, using for the piffpose Eucl. i. 13, 15 and 16 (or
18). The Diodorus here mentioned may be the author of the Analemma on
which Pappus wrote a commentary. It is difficult even to frame a conjecture
as to who " Abthiniathus " is. In one place in the Arabic text the name
appears to be written " Anthisathus " (H. Suter in Zeitschrift fiir Math, und
Pkysiky xxxviii.y hist, litt Abth. p. 194). It has occurred to me whether he
might be Peithon, a friend of Serenus of Antinoeia (Antinoupolis) who was
long known as Serenus of Antissa, Serenus says (De sectione cylindri^ ed.
Heiberg, p. 96): "Peithon the geometer, explaining parallels in a work of his,
was not satisfied with what Euclid said, but showed their nature more cleverly
by an example; for he says that parallel straight lines are such a thing as we
see on walls or on the ground in the shadows of pillars which are made when
either a torch or a lamp is burning behind them. And, althougn this has only
been matter of merriment to every one, I at least must not deride it, for the
respect I have for the author, who is my friend.'' If Peithon was known as
" of Antinoeia " or " of Antissa,** the two forms of the mysterious name might
perhaps be an attempt at an equivalent; but this is no more than a guess.

304 BOOK I [i. Post. 5

Simplicius adds in full and word for word the attempt of his "friend** or
his *' master Aganis " to prove the Postulate.

Proclus returns to the subject (p. 365, 5) in his note on EucL i. 39. He
says that before his time a certain numb^ of g^meters had classed as a
theorem this Euclidean postulate and thought it matter for proof, and he then
proceeds to give an account of Ptolem/s argument

Noteworthy attempts to prove the Postulate.

Ptolemy.

We learn from Proclus (p. 365, 7 — 11) that Ptolemy wrote a book on the
proposition that " straight lines drawn from angles less than two right angles
meet if produced," and that he used in his ''proof" many of the theorems in
Euclid preceding i. 29. Proclus excuses himself from reproducing the nrly
part of Ptolemy's argument, only mentioning as one of the propositions
proved in it the theorem of EucL i. a8 that, if two straight lines meeting a
transversal make the two interior angles on the same side equal to two ri^t
angles, the straight lines do not meet, however far produced.

I. From Proclus' note on 1. 28 (p. 362, 14 sq.) we know that Ptdemy
proved this somewhat as follows.

Suppose that there are two straight lines AB, CD, and that EFGH,
meeting them, makes the angles BFG, FGD equal to two right angles.
I say that AB, CD are paralH that is, they
are non-secant

For, if possible, let FB, GD meet at K.

Now, since the angles BFG, FGD are
equal to two right angles, while the four
angles AFG, BFG, FGD, FGC are together
equal to four right angles,

the angles AFG, FGC are equal to two
right angles.

^^If therefore FB, GD, when the interior angles are equal to two right
angles, meet at K, the straight lines FA, GC will also meet if produced; for the
angles AFG, CGFaie also equal to two right angles.

''Therefore the straight lines will either meet in both directions or in
neither direction, if the two pairs of interior angles are both equal to two right
angles.

" Let, then, FA, GC meet at Z.

"Therefore the straight lines LABK^ LCDK enclose a space : which is
impossible.

"Therefore it is not possible for two straight lines to meet when the
interior angles are e^ual to two right aisles. Therefore they are parallel"

[The argument m the words italic^ed would be clearer if it had been
shown that the two interior angles on one side of EH are severally equal to the
two interior angles on the other, namely BFG to CGF and FGD to AFG\
whence, assuming FB, GD to meet in K, we can take the triangle KFG and
place it (e.g. by rotating it in the plane about O the middle point of FG) so
diat FG falls where GF\& in the figure and GD fdls on FA, in which case
FB must also frdl on GC\ hence, since FB, GD meet at K, GC and FA
must meet at a corresponding point Z. Or, as Mr Frankland does, we may
substitute for FG a straight line MN through O the middle point of FG
drawn peipendicular to one of the parallels, say AB, Then, smce the two
triangles OMF, ONG have two angles equal respectively, namely FOM to

r

z

1. Post. 5] NOTE ON POSTULATE 5 . 205

GON(i. is) and OFMio OGN, and one side O-^ equal to one side OG^ the
triangles are congruent, the angle ONG is a right angle, and MN is perpen-
dicular to both AB and CD. Then, by the same method of application,
MA^ NC are shown to form with MN di triangle MALCN congruent with
the triangle NDKBM, and MA^ NC meet at a point L corresponding to K.
Thus the two straight lines would meet at the two points K^ JL This is what
happens under the Riemann hypothesis, where the axiom that two straight
lines cannot enclose a space does not hold, but all straight lines meeting in
one point have another point common also, and e.g. in the particular figure
just used K^ L are pomts common to all perpendiculars to MN If we
suppose that K^ L are not distinct points, but om point, the axiom that two
straight lines cannot enclose a space is not contradicted.]

II. Ptolemy now tries to prove i. 29 without using our Postulate, and
tl.wii deduces the Postulate from it (Proclus, pp. 365, 14 — 367, 27).

The argument to prove i. 29 is as follows.

The straight line which cuts the parallels must make the sum of the
interior angles on the same side equal to, greater
than, or less than, two right angles. ^ ^ ?

"Let AB, CD be parallel, and let FG meet
them. I say (i) that FG does not make the
interior angles on the same side greater than two g —
right angles.

" For, if the angles AFG, CGF are greater than two right angles, the
remaining angles BFG, DGF are less than two right angles.

*' But the same two angles are also greater than two right angles ; for AF,
CG are no more parallel than FB, GD, so that, if the straight line falling on
AF, CG makes the interior angles greater than two right angles, the straight line
falling on FB, GD will also make the interior angles greater than two right
angles.
V " But the same angles are also less than two right angles ; for the four
angles AFG, CGF, BFG, DGF are equal to four right angles :
which is impossible.

"Similarly (2) we can show that the straight line falling on the parallels
does not make the interior angles on the same side less than two right angles.

" But (3), if it makes them neither greater nor less than two right angles,
it can only make the interior angles on the same side equal to two right
angles.*'

III. Ptolemy deduces Post. 5 thus:

Suppose that the straight lines making angles with a transversal less than
two right angles do not meet on the side on which those angles are.

Then, a fortiori, they will not meet on the other side on which are the
angles greater than two right angles.

Hence the straiglit lines will not meet in either direction ; they are there-
fore parallel.

But, if so, the angles made by them with the transversal are equal to two
right angles, by the preceding proposition (= i. 29).

Thaj^fore the same angles will be both equal to and less than two right
angles : ▼
which is impossible.

Hence the straight lines will meet

9o6

BOOK I

[l PdST. 5

IV. Ptolemy lastly enforces his condusion that the strai^t lines will
meet on the side on which are the emgUs less than two right am^ by recurring
to the ajbrtiofi step in the forgoing proof.

Let the angles AFG^ CGJF'm the accompanying figure be together less
than two right angles. '

Therefore the angles BFG^ DGF are greater
than two right angles.

We have proved that the straight lines are not
non-secant

If they meet, they must meet either towards
A^ C or towards B^ D.

(i) Suppose they meet towards B^ A at K.

Then, smce the angles AFG^ CGFwm less than
two right angles, and the angles AFG^ GFB are
equal to two right angles, take away the common angle AFG, and

the angle CGF is less than the angle BFG;

that is^ the exterior angle of the triangle JI^FG is less than the interior and
opposite angle BFG :
which is impossible.

Therefore AB, CD do not meet towards B^ D.

(2) But they do meet, and therefore they must meet in one direction or
the other :

therefore they meet towards A^ B^ that is, on the side where are the
angles less than two right angles.

The flaw in Rolemy's argument is of course in the part of his proof of
I. 29 which I have italicised As Produs says, he is not entitled to iissume
that, if AB^ CD are parallel, whatever is true of the interior angles on one
side of FG (i.e. that they are together equal to, greater than, or less than, two
right angles) is necessarily true at the same time of the interior angles on the '
other side. Ptolemy justifies this by saying that FA^ GC are no more paralld %
in one direction tlum FB^ GD are in the other : which is equivalent to the i
assumption that through any point only one parallel can be drawn to a given ,
straight line. That is, he assumes an equivalent of the very Postulate he b :
endeavouring to prove. ^

Proclus.

I

Before passing to his own attempt at a proof, Proclus (p. 368, 26 sqq.) ^
examines an ingenious argument (recalling somewhat the famous one about \
Achilles and the tortoise) which appeared to show that it was impossible for -
the lines described in the Postulate to meet j

Let AB, CD make with ^C the angles BAC, A CD together less than ^
two right angles.

Bisect AC at E and along AB, CD
respectively measure AF^ CG so that each
is equal to AE.

Bisect FG at H and mark off FK,
GL each equal to FH\ and so on.

Then AF^ CG will not meet at any
point on FG ; for, if that were the case, two sides of a triangle would be
together equal to the third: which is impossible.

e} \ -h

fc— 5

r

I. Post, s] NOTE OK POSTULATE 5 ao?

Simflarly, AB^ CD will not meet at any point on KL \ and ''proceeding
like this indefinitely, joining the non-coincident points, bisecting the lines so
drawn, and cutting off from the straight lines portions equal to the half of
these, they say they thereby prove tlmt the straight lines AB^ CD will not
meet anywhere."

It is not surprising that Proclus does not succeed in exposing the fallacy
here (the fact bemg that the process will indeed be endless, and yet the straight
lines will intersect within a finite distance). But Proclus' criticism contains
nevertheless something of value. He says that the argument will prove too
much, since we have only to join AGva order to see that straight lines making
some angles which are together less than two right angles do in fact meet,
namely AG^ CO. "Therefore it is not possible to assert, without some definite
limitation, that the straight lines produced from angles less than two right
angles do not meet On the contrary, it is manifest that same straight lines,
when produced from angles less than two right angles, do meet, although the
argument seems to require it to be proved that this property belongs to aU
such straight lines. For one might say that, the lessening of the two right
angles being subject to no limitation, with such and such an amount of
lessening the straight lines remain non-secant^ but with an amount of lessening
in excess of this they meet (p. 371, 2—10)."

[Here then we have the germ of such an idea as that worked out by
Lobachewsky, namely that the straight lines issuing from a point in a plane
can be divided with reference to a straight line lying in that plane into two
classes, ''secant" and "non-secant," and that we may define as parallel the
two straight lines which divide the secant from the non-secant class.]

Proclus goes on (p. 371, 10) to base his own argument upon "an axiom
such as Aristotle too used in arguing that the universe is finite. -For, if from
one point two straight lines forming an angle Ife produced indefinitely^ the distance
(&acrTao'c¥, Arist. Scoon/fui) between the said straight lines produ^ indefinitely
will exceed any finite magnitude. Aristotle at all events showed that, if the
straight lines drawn from the centre to the circumference are infinite, the
interval between tliem is infinite. For, if it is finite, it is impossible to
increase the distance, so that the straight lines (the radii) are not infinite.
Hence the straight lines, when produced indefinitely, will be at a distance from
one another greater than any assumed finite magnitude."

This is a fair representation of Aristotle's argument in De caeh 1. 5, 271
b 28, although of course it is not a proof of what Proclus assumes as an
axiom.

This being premised, Proclus proceeds (p. 371, 24) :

I. " I say that, if any straight line cuts one of two parallels^ it will cut
\ the other also,

"For let AB, CD be parallel, and let EFG cut AB\ I say that it will cut
k CD also.
' " For, since BF^ FG are two straight lines from ^

one point F^ they have, when produced indefinitely, ^ X^ g

a distance greater than any magnitude, so that it will ^\

J also be greater than the interval between the parallels. Q

' Whenever therefore they are at a distance from one ^ q

another greater than the distance between the parallels,
FG wiU cut CD.
" Therefore etc"

ao8 BOOK I

J

[l Post. 5 I

II. *' Having proved this, we shall prove, as a deduction firom it, the
theorem in question.

'' For let AB^ CD be two straight lines, and let £^ (idling on them make
the angles BEF^ DFE less than two right angles.

''I say that the straight lines will meet on that
side on which are the angles less than two right
angles.

" For, since the angles BEF^ DFE are less
than two right angles, let the angle HEB be equal
to the excess of two right angles (over them), and let HE be produced to K.

''Since then EF falls on KH^ CD and makes the two interior ang^
HEF^ DFE equal to two right angles,

the straight lines HK^ CD are parallel.

'' And AB cuts KH\ therefore it will also cut CZ>, by what was before
shown.

''Therefore AB^ CD will meet on that side on which are the angles less
than two right angles.

"Hence the theorem is proved."

Clavius criticised this proof on the ground that the asdom from which
it starts, taken from Aristotle, itself requires proofl He points out diat, just
as you cannot assume that two lines which continually approach one another
will meet (witness the hyperbola and its asymptote), so you cannot assume
that two lines which continually diverge will ultimately be so fiLr apart that a
perpendicular from a point on one let fall on the other will be greater than
any assigned distance ; and he refers to the conchoid of Nicororiies, whidi
continually approaches its asymptote, and therefore continually gets fartfier
away from the tangent at the vertex ; yet the perpendicular from any point on
the curve to that tangent will always be less than the distance between the
tangent and the asymptote. Saccheri supports the objection.

Proclus' first proposition is open to the objection that it assumes that two
"parallels" (in the Euclidean sense) or, as we may say, two straight lines
which have a common perpendicular, are (not necessarily equidistant, but)
so related that, when they are produced indefinitely, the perpendicular from a
point of one upon the other remains finite.

This last assumption is incorrect on the hyperbolic hypothesis; the
"axiom" taken from Aristotle does not hold on the elliptic hypothesis.

Na^iraddln at-T^si.

The Persian-bom editor of Euclid, whose date is 1 201— 1274, has three
lemmas leading up to the final proposition. Their content is substantially as
follows, the first lemma being apparently assumed as evident

I. (a) If AB^ CD be two straight lines such that successive perpen-
diculars, as EF^ GH^ KL^ from points on AB to CD always make with AB
unequal angles, which are always acute on the side towards B and always
obtuse on Sie side towards A^ then the lines AB^
CD^ so long as they do not cut, approach continually
nearer in the direction of the acute angles and diverge
continually in the direction of the obtuse angles, and
the perpendiculars diminish towards B^ D^ and in-
crease towards A^ C. 5 — [^ |!| ( r

ip) Conversely, if the perpendiculars so drawn
continually become shorter in the direction of B^ Z>, and longer in the

I. Post. 5]

NOTE ON POSTULATE 5

909

direction of A^ Q the straight lines AB^ CD approach continually nearer in
the direction of B^ D and diverge continually in the other direction ; also
each perpendicular will make with AB two angles one of which is acute and
the other is obtuse, and all the acute angles will lie in the direction towards
B^ D^ and the obtuse angles in the opposite direction.

[Saccheri points out that even the first part (a) requires proof. As
regajrds the converse (p) he asks, why should not the successive acute angles
made by the perpendiculars with AB^ while remaining acute, become greater
and greater as the perpendiculars become smaller until we arrive at last at a
perpendicular which is a common perpendicular to both lines? If that happens,
all the author's efforts are in vain. And, if you are to assume the truth of the
statement in the lemma without proof, would it not, as Wallis said, be as
easy to assume as axiomatic the statement in Post 5 without more ado?]

II. ^ AC, BD he drawn from the extremities of AB at right angles to it
and on the same side^ and if AC, BD be made equal to one another and CD be

joined^ each of the angles ACD, BDC will be rights and
CD will be equal to AB.

The first part of this lemma is proved by reductio ad
absurdum from the preceding lemma. If, e.g., the angle
ACD is not right, it must eitiier be acute or obtuse.

Suppose it is acute; then, by lemma i, ^^C is greater
than BD^ which is contrary to the hypothesis. And so on.

The angles ACD^ BDC being proved to be right angles, it is easy to
prove that AB^ CD are equal.

[It is of course assumed in this "proof" that, if the angle ACD is acute,
the angle BDC is obtuse, and vice versa.]

III. /n any triangle the three angles are together equal to two right angles.
This is proved for a right-angled triangle by means of the forgoing lemma,

the four angles of the quadrilateral ABCD of that lemma being all right angles.
The proposition is then true for any triangle, since any triangle can be divided
into two right-angled triangles.

IV. Here we have the final "proof" Of Post. 5, Three cases are
distinguished, but it is enough to show the case where one of the interior
angles is right and the other acute.

Suppose AB^ CD to be two straight lines met by FCE making the angle
ECD a right angle and the angle CEB
an acute angle.

Take any point G on £B^ and draw
GJ£ perpendicular to EC

Since the angle CEG is acute, the
perpendicular GH will fall on the side of
E towards 2?, and will either coincide
with CD or not coincide with it In the
former case the proposition is proved.

If GH does not coincide with CD
but falls on the side of it towards /^ CZ>, being within the triangle formed by
the perpendicular and by C£, EG^ must cut EG. [An axiom is here used,
namely that, if CD be produced far enough, it must pass outside the triangle
and therefore cut some side, which must be EB^ since it cannot be the
perpendicular (i. 27), or CE,]

Lastly, let GHtaM on the side of CD towards E.
H. B. 14

9IO BOOK I [l Post. 5

Along JTCset of[ HK, KL etc, etch equal to EH^ iintU we get the fint
point of division, as M^ beyond C

Along GB set off GN^ NO etc, each equal to BG^ until EP ii the mne
multiple of EG that ^ATis of EH.

llien we can prove that the perpendiculars from N^ O^ P on EC fiUl on
the points K^ Z, AT respectively.

For take the first perpendicular, that from N^ and call it N&

Draw EQ at right angles to Elf and equal to GB^ and set off 5!i? along
5A^also equal to GIf. Join Q(r, GE.

Then (second lemma) the angles EQGf QGIfare right, and QG « Elf.

Similarly the angles SEG, RGffaxt right, and RG^SH.

Thus EGQ is one straight line, and the vertically opposite angles NGR^
EGQ are equal The angles NRG^ EQG are both right, and NG*^ GE^ by
construction.

Therefore (i. 26) EG = GQ;

whence SIf=^ HE « KH^ and 5 coincides with K.

We may proceed similarly with tlie other perpendiculars.

Thus PM is perpendicular to FE. Hence CD^ being parallel to MP and
within the triangle PME^ must cut EP^ if produced f»x enough.

John Wallis.

As is well known, the argument of Wallis (16x6 — 1703) assumed as a
postulate that, given a figure^ another figure is possible which is similar io the
gitfen one and of any size whatever. In fiEurt Wdlis assumed this for irbmgles
only. He first proved (i) that, if a finite straight line is placed on an infinite
straight line, and is then moved in its own direction as far as we please^
it will always lie on the same infinite straight line, (2) that, if an ai^jle be
moved so that one 1^ always slides along an infinite straight line^ the angle
will remain the same, or equal, (3) that, if two straight lines, cut by a third,
make the interior angles on the same side less than two right angles, each
of the exterior angles is greater than the opposite
interior angle (proved by means of i. 13).

(4) If AB, CD make, with A C, the interior
angles less than two right angles, suppose AC
(with AB rigidly attached to it) to move along
AF to the position ay, such that a coincides
with C If AB then takes the position aj3, a^ lies entirely outside CD (proved
by means of (3) above).

(5) With the same hypotheses, the straight line a^, or AB, during its
motion^ and before a reaches C, must cut the straight line CD.

!6) Here is enunciated the postulate stated above.
7) Postulate 5 is now proved thus.

Let AB^ CD be the straight lines which make, with the infinite straight
line ACF meeting them, the interior angles
BA C, DC A together less than two right angles.

Suppose AC (with AB ri^dly attached to
it) to move along ACF until AB takes the
position of o^ cutting CD in ir.

Then, aCW being a triangle, we can, by
the above postulate, suppose a triangle drawn
on the base CA similar to the triangle aCif.

Let it be ACP.

[Wallis here interposes a defence of the hypothetical construction.]

1

I. Post. 5] NOTE ON POSTULATE s aii

Thus CP and AP meet at P\ and« as by the definition of similar figures
the angles of the triangles PCA^ wCa, are respectively equal, the angle PC A
being equal to the angle irCa and the angle PAC to the angle waC or BAC^
it follows that CP^ AP lie on CD^ AB produced respectively.

Hence AB^ CD meet on the side on which are the angles less than two
right angles.

[The whole gist of this proof lies in the assumed postulate as to the
existence of similar figures ; and« as Saccheri points out, this is equivalent to
unconditionally assuming the ''hypothesis of the right angle," and consequently
Euclid's Postulate 5.]

Gerolamo Saccheri.

The book EucUdes etb omni ncuvo vindicaius (1733) by Gerolamo Saccheri
(1667 — 1733), a Jesuit, and professor at the University of Pavia, is now
accessible (i) edited in German by Engel and Stiickel, DU Theork der
ParaUtllinien von Euklid bis auf Gauss^ 1895, PP- 4' — ^3^9 ^"^ (?) ^^ ^^

(Italian version, abridged but annotated, LEudide em€nd€Uo del P, Gerolamo
Saccheri^ by G. Boccardini ^Hoepli, Milan, 1904). It is of much greater |
importance than all the earlier attempts to prove Post 5 because Saccheri
was the first to contemplate the possibility of hypotheses other than that of
y Euclid, and to work out a number of consequences of those hypotheses.
He was therefore a true precursor of Legendre and of Lobachewsky, as
Beltrami called him (1889), and, it might be added, of Riemann also. For,
as Veronese observes (Fondamenii di ^ometria^ p. 570), Saccheri obtained
a glimpse of the theory of parallels m all its generality, while L^endre,
Lobachewsky and G. Bolyai excluded a /r/V^r/, without knowing it, the ''hypo-
thesis of the obtuse angle, '^ or the Riemann hypothesis. Saccheri, however,
was the victim of the preconceived notion of his time that the sole possible
/ geometry was the Euclidean, and he presents the curious spectacle of a man
, Uiboriously erecting a structure upon new foundations for the very purpose of
\ demolishing it afterwards ; he sought for Contradictions in the heart of the
I systems which he constructed, in order to prove thereby the falsity of his
hypotheses.

For the purpose of formulating his hypotheses he takes a plane quadri-
lateral ABDCy two opposite sides of wluch, AC^ BD^
are equal and perpendicular to a third AB. Then the
angles at C and D are easily proved to be equal On
the Euclidean hypothesis they are both right angles;
but apart from this hypothesis they might be both
obtuse, or both acute. To the three possibilities, which
I Saccheri distinguishes by the names (i) the hypothesis of
the right angle^ (2) the hypothesis of the obtuse angle and
t (3) the hypothesis of the acute angle respectively, there corresponds a certain
group of theorems ; and Saccheri's point of view is that the Postulate will
be completely proved if the consequences which follow from the last two
hypotheses comprise results inconsistent with one another.

Amons the most important of his propositions are the following :
(i) Ijthe hypothesis of the right angle^ or of the obtuse an^^ or of the acute
am^ is proved true in a single case, it is true in ettery other case. (Props, v.,

VI., Vll.)

(a) According as the hypothesis of the right angle, the obtuse angle, or the
acute angle is true, the sum of the three .angles of a triangle is equal to, greater
than, or less than two right angles. (Prop, ix.)

14—2

C

"•-.

/^<.

A i

I

fx2 BOOK I [lPoct. 5

(3) From thi exUtence of a sitigk irian^ in tMth ih$ sum of the angUs is
equal to^ gnaUr ihan^ or less ikon two right augles tki truth of the hypothesis
of the right angle^ obtuse angle^ or acute em^ respeetivefy fottoms. (Pi^ xv.)

These propositions involve the foUowing \ If in a single triangle the sum
of the angles is equal to, greater than^ or less them two right angUs^ then any
triangle has the sum of its angles equal to^ greater thau^ or less than two right
angles respectively^ which was (voyed about a century later by Lq;endre for
the two cases only where the sum is equal to or less than two right angles.

The proofs are not free from imperfections, as when, in the poob of
Prop. XII. and the part of Prop. xiii. relating to the hypothesis of the obtuse u
angle, Saccheri uses Eucl i. t8 depending on l 16, a proposition which is |(
only valid on the assumption that straight lines are infinite tn length ; for this 1 1
assumption itself does not bold under the hypothesis of the obtuse angle
(the Riemann hypothesis).

The hypothesis of the acute angle takes Saccheri much longer to dispose
of, and this part of the book is less satisfactory ; but it contains die following
propositions afterwards established anew by Lobachewsky and Bolyai, viz. :

(4) Two straight lines in a plane (even on the hypothesis of the acute
angle) either have a common perpendieular, or must, if produced in one and the
same direction, either intersect once at a finite distance or at least continueUfy
approach one another, (Prop, xxiii.)

(5) In a cluster of rays issuing from a point there exist aluHxys (on the
hypothesis of the acute angle) two determinate straight lines which sej^araie the
straight lines which intersect a fixed straight line from those whuh do not .
intersect it, ending with and including the straight line which has a common
perpendicular with the fixed straight line. (Props, xxx., xxxi., xxxii.)

Lambert.

A dissertation by G.S. Kliigel, Conatuum praecipuorum theoriam parallelarum
demonstrandi recensio (i 763), contained an examination of some thirty '* demon-
strations" of Post 5 and is remarkable for its conclusion expressing, apparently
for the first time, doubt as to its demonstrability and observing that the \
certainty which we have in us of the truth of the Euclidean hypothesis is «
not the result of a series of rigorous deductions but rather of experimental <
observations. It also had the greater merit that it called the attention of 1
Johann Heinrich Lambert (1728 — 1777) to the theory of parallels. His
Theory of Parallels was vrritten in 1766 and published after his death by
G. Bernoulli and C. F. Hindenburg ; it is reprcxluced by Engel and Stackel !
(op. cit. pp. 152 — 208). y

The third part of Lambert's tract is devoted to the discussion of the same |
three hypotheses as Saccheri's, the hypothesis of the right an^ being for
Lambert the first, that of the obtuse angle the second, and that of the acute ]
angle the thirds hypothesis; and, with reference to a quadrilateral with three '
right angles from which Lambert starts (that is, one of the halves into which
the median divides Saccheri's quadrilateral), Uie three hypotheses are the i
assumptions that the fourth angle is a right angle, an obtuse angle, or an '
acute angle respectively.

Lambert goes much further than Saccheri in the deduction of new \
propositions from the second and third hypotheses. The roost remarkable is {
the following. ,

The area of a plane triangle^ under the second and third hypotheses, is
proportional to the eUfference between the sum of the three cmgla and two right
angles.

1

i

L Post. $] NOTE ON POSTULATE $ ax3

Thus the numerical expression for the area of a triangle is, under the
I Mrd hypothesis

t £L^k\w^A^B^C) (i).

I and under the sicond hypothesis

I A = *(^+i?+C-») (a).

' where ^ is a positive constant

A remarkable observation is appended (§ 82) : '' In connexion with this it
seems to be remarkable that the second hypothesis holds if spherical instead of
plane triangles are taken, because in the former also the sum of the angles is
gieater than two right angles, and the excess is proportional to the area of the
triangle.

''It appears still more remarkable that what I here assert of spherical
triangles can be proved independently of the difficulty of parallels.'

This discovery that the second hypothesis is realised on the surface of a
sphere is important in view of the development, later, of the Riemann
hypothesis (1854).

Still more remarkable is the following prophetic sentence: **Iam almost
inclined to draw the conclusion that the third hypothesis arises with an imaginary
spherical surfau^^ {ci, Lobachewsky's Gkomitrie ima^naire^ 1^37)*
^ No doubt Lambert was confirmed in this by the feurt that, in the formula

(2) above, which, for ^ = f', represents the area of a spherical triangle, if
r V- I is substituted for r, and r^=^h^ we obtain the formula (i).

Legendre.

No account of our present subject would be complete without a full
reference to what is of permanent value in the investigations of Adrien Marie
L^endre (i 752-71833) relating to the theory of parallels, which extended over
, the space of a generation. His different attempts to Drove the Euclidean
\ hypothesis appeared in the successive editions of his Elkments de Ghmitrie
I ^m the first (1794) to the twelfth (1823), which last may be said to contain

I his last word on the subject. Later, in 1833, he published, in the Mhnoires
de rAcadtmie Royale des Sciences^ xii. p. 367 sqq., a collection of his different
proofs under the title Rhflexions sur aijjfirentes mani^res de dhnontrer la thhrie
\ des paraUkks. His exposition brought out clearly, as Saccheri had done, and
kept steadily in view, the essential connexion between the theory of parallels
and the sum of the angles of a triangle. In the first edition of the Elkments
the proposition that the sum of the angles of a triangle is equal to two right
angUs was proved analytically on the basis of the assumption that the choice
of a unit of length does not affect the correctness of the proposition to be
proved, which is of course equivalent to Wallis' assumption of the existence of
similar figures, A similar analytical proof is given in the notes to the twelfth
edition. In his second edition L^endre proved Postulate 5 by means of the
assumption that, ^ven three points not in a straight line^ there exists a arcle
passing through cUl three. In the third edition (1800) he gave the proposition
that the sum of the an^s of a triangle is not greater than two ri^ angles]
this proof, which was geometrical, was replaced later by another, the best
known, depending on a construction like that of Euclid 1. 16, the continued
application of which enables any number of successive triangles to be evolved
in which, while the sum of the angles in each remains always equal to the
sum of the angles of the original triangle, one of the angles increases and the
sum of the other two diminishes continually. But Legendre found the proof
of the equally necessary proposition that the sum of the angles of a triangle is

JI4 BOOK I [l FdiT. 5

not less than two right angles to present great diflfculties. He first observed
that, as in the case of spherical trian^es (in whidi the sum of the angles it
greater than. two right angles) the excess of the sum of the apgles over two
right angles is proi)ortional to the area of the triangle, so in the case of
rectilineal triangles, if the sum of the angles is less than two right angles by «
certain defidiy th^ deficit will be pioportbnal to the area of the trian^
Hence if, starting from a given triangle, we could construct another triaiu^
in which the original triangle is contained at least m times, the d^icU of mis
new triangle will be equal to at least m times that of the original trian^^ so
that the sum of the angles of the greater triang^ will diminish pro g re s s i vdy
as m increases, until it becomes lero or negative: which is absurd. The
whole difficulty was thus reduced to that of the construction of a trianrie
containing the given triangle at least twice; but the solution of even mis
simple problem requires it to be assumed (or proved) that tknm^ a ghm
/0iftt within a given angle less than two4hirds of a ri^ angle we eon eihoays
draw a straight tine which shall meet both sides of the angle. This is however
really equivalent to Euclid's Postulate. The proof in the course of whidi the
necessity for the assumption appeared is as fbUows.

It is required to prove that the sum of the angles of a triai^ cannot be
less than two right angles.

Suppose A is the least of the three angles of a triangle ABC. Apply to
the oj^posite side BC a triangle DBQ eqiud to
the triangle ACB, and suoi that the angle
DBC is equal to the angle ACB^ and the angle
DCB to the angle ABC ; and draw any straight
line through D cutting AB^ AC produced in

If now the sum of the angles of the triangle f
ABC is less than two right angles, being equal

to 2^-S say, the sum of the angles of the triangle DBC^ equal to the
triangle ABC, is also 2^-S. \

Smce the sum of the three angles of the remaining triangles DEB, FDC I
respectively cannot at all events be greater than two right angles [for L^endre's |
proofs of this see below], the sum of the twelve angles of tibe four triangles in i
the figure cannot be greater than I

4^ + (a^««) + (2/?-«X >e. 8^-28.

Now the sum of the three angles at each of the points B^ C, Dv&iR. i

Subtracting these nine angles, we have the result that the three angles of '
the triangle AEF cannot be greater than 2^-2$. j

Hence, if the sum of the angles of the triangle ABC is less than two right ^
angles by h, the sum of the angles of the laiger triangle AEEis less than two ^
right angles by at least 2S. ,

We can continue the construction, making a still larger triangle firom AEE, [
and so on.

But, however small 8 is, we can arrive at a multiple 2*8 which shall exceed :
any given angle and therefore 2B itself; so that the sum of the three angles I
of a triangle sufficiently laige would be zero or even less than zero : which is
absurd.

Therefore etc.

The difficulty caused by the necessity of making the above-mentioned
assumption made Legendre abandon, in his ninth edition, the inethod of the

1

t

I. Post, s] NOTE ON POSTULATE 5 215

editions from the third to the eighth and return to Euclid's method pure and
simple.
) But again, in the twelfth, he returned to the plan of constructing any

I number of successive triangles such that the sum of the three angles in all of
I them remains equal to the sum of the three angles of the original triangle,
but two of the angles of the new triangles become smaller and smaller, while
the third becomes larger and laiger ; and this time he claims to prove in one
proposition that the sum of the three angles of the original triangle is r^a/ to
two right angles by continuing the construction of new triangles indefinitely
and compressing the two smaller angles of the ultimate triangle into nothing,
while the third angle is made to become a flat angle at the same time. The
construction and attempted proof are as follows.

Let ABC be the given triangle ; let AB be the greatest side and BC the
least ; therefore C is the greatest angle and A the least.

From A draw AD to the middle point of BC^ and produce AD to C",
making AC equal to AB.

Produce AB to B^ making AB equal to twice AD.

The triangle ABC is then such that the sum of its three angles is equal
to the sum of the three angles of the triangle ABC.

For take ^ A' along AB equal to AD^ and join CK.

Then the triangles ABD^ ACK have two sides and the included angles
respectively equal, and are therefore equal in all respects ; and CK is equal to
BD or DC

Next, in the triangles BCK^ ACD, the angles BKC\ ADC are equal,
being respectively supplementary to the equal angles AKC\ ADB\ and the
two sides about the equal angles are respectively equal;

therefore the triangles BCK^ A CD are equal in all respects.

Thus the angle ACB is the sum of two angles respectively equal to the
angles B^ C of &e original triangle ; and the angle A in the original triangle
is the sum of two angles respectively equal to the angles at A and B in the
triangle ABC.

It follows that the sum of the three angles of the new triangle ABC is
equal to the sum of the angles of the triangle ABC.

Moreover, the side AC\ bein^ equal to AB^ and therefore greater than
AC^ is greater than BC which is equal to AC.

Hence the angle CAB is less than the angle ABC ; so that the angle
CAB is less than ^A^ where A denotes the angle CAB of the original
triangle.

[It will be observed that the triangle ABC is really the same triangle as
the triangle AEB obtained by the construction of Eucl. i. 16, but differendy
placed so that the longest side k'es along AB.]

By taking the midcUe point D of the side B'C and repeating the same
construction, we obtain a triangle A B'C' such that (i) the sum of its three
angles is equal to the sum of the three angles of ABCf (2) the sum of the

I

9i6 BOOK I [lFobt. 5

two angles C'AB\ AB'C* is equal to the aii|^ CAB in tbe preceding
triangle, and is therefore less than \A^ and (3) the an|^ CAB* is kas than
half &e angle CAB>^ and therefore ie» than ^A.

Continuing in this way, we shall obtain a triangle Ahc such that the sum of

two angles, those at A and b^ is less than -^A^ and the an|^ at r is greater

than the corresponding angle in the preceding triangle.

If, L^endre argues, the construction be continued indefinitely so that

-^A becomes smaller than any assigned angles the point c ultimately lies on

Aby and the sum of the three ai^es of the triangle (which is equal to the sum j
of the three angles of the original triangle) becomes identical with the angle |
at r, which is then 2^ flat angle, and therdrore equal to two right angles.

This proof was however shown to be unsound (in remct Si the final K
inference) by J. P. W. Stein in Gergonne's Afmales de MoiUmaHfuis xv.,
1824, pp. 77—79-

We will now reproduce shortly the substance of the theorems of Legendre
which are of the most permanent value as not depending on a particular
hypothesis as regards parallels.

I. 7%e sum of the thre^ anf^ cf a irian^ cannci be greater than iW0
right atigks.

This Legendre proved in two ways.

(i) First proof (m the third edition of the iUments).

Let ABC be the given triangle, and ACJ9l straight line.

Make CE equal to AC, the angle DCE equal to the angle BAC^ and DC
equal to AB. Join DE.

Then the triangle DCE is equal to the triangle BAC in all respects.

If then the sum of the three angles of the triangle ABC is greater than

{

{

2Ey the said sum must be greater than the sum of the angles BCA^ BCD,
DCE, which sum is e^al to 2^.

Subtracting the equal angles on both sides, we have the result that i

the angle ABC is greater than the angle BCD. (

But the two sides AB, BC of the triangle ABC are respectively equal to i
the two sides DC, CB of the triangle BCD.

Therefore the base AC is greater than the base BD (EucL i. 24).

Next, make the triangle FEG (by the same construction) equal in all ^
respects to the triangle BAC or DCE; and we prove in the same way that *
CE (or AC) is greater than DF.

And, at the same time, BD is equal to DF^ because the angles BCD,
DEFMxe equal.

Continuing the construction of fiirther triangles, however small the
difference between AC and BD is, we shall ultimately reach some multiple j

' 1.P0ST. s] NOTE ON POSTULATE $ »i7

I of this difference, represented in the figure by (say) the difference between
^. the straight line A/ and the composite line BDFHK^ which will be greater
than any assigned length, and greater therefore than the sum of AB vcAJK,

Hence, on the assumption that the sum of the angles of the triangle ABC
is greater than 2^, the broken line ABDFHKJ may be less than the straight
line AJ\ which is impossible.

Therefore etc

(2) Proof substituted later.

If possible^ let 2^ -f a be the sum of the three angles of the triangle ABC^
of which A is not greater than either of the
others.

Bisect BC at H^ and produce AH to Z>,
making HD equal to AH\ join BD.

Then the triangles AHC^ DHB are equal in
all respects (i. 4) ; and the angles CAH^ ACHzx^
respectively equal to the angles BDH^ DBH.

It follows that the sum of the angles of the
triangle ABD is equal to the sum of the angles of the original triangle, i.e.
to 2^ + a.

And one of the angles DAB^ ADB is either equal to or less than half the
angle CAB.

Continuing the same construction with the triangle ADB^ we find a third
triangle in which the sum of the angles is still 2^ + 0, while one of them is
equal to or less than ( l CAB)I^.

Proceeding in this way, we arrive at a triangle in which the sum of the
angles is 2^ + a, and one of them is not greater than ( l CAB)l2\

And, if n is sufficiently large, this will be less than a ; in which case we
should have a triangle in which two angles are together greater than two right
angles : which is absurd.

Therefore a is equal to or less than zero.

(It will be noted that in both these proofs, as in Eucl. i. 16, it is taken for
granted that a straight One is infinite in length and does not return into itself,
which is not true under the Riemann hypothesis.)

II. On the assumption that the sum of the angles of a triangle is less
than two right angles, if a triangle is made up of two others^ the ^^ deficit** of the
former is equal to the sum of the ^^ deficits** of the others.

In fact, if the sums of the angles of the component triangles are 2^ - a,
i/i-p respectively, the sum of the angles of the whole triangle is

(2^-a) + (2-^-/9)-2^ = 2-^-(a + /9).

III. If the sum of the three angles of a triangle is equal to two ri^t
angles^ the same is true of all trian^es obtained by subdividing it by straight
lines drawn from a vertex to meet the opposite side.

Since the sum of the angles of the triangle ABC is equal to 2^, if the
sum of the andes of the triangle ABD were 2^- a, it
would follow that the sum of the angles of the triangle
ADC must be 2^ + a, which is absurd (by I. above).

IV. If in a triangle the sum of the three angles is
equal to two right angles ^ a quadrilcUeral can cUways be
constructed with four right angles and four equal sides
exceeding in length any assigned rectilineal segment.

Let ABC be a triangle in which the sum of the angles is equal to two

si8

BOOK I

[lFobt. 3

Tight angles. We can assume ABC to be an isaseeies rigfit-am^ki triaiude
because we can reduce the case to this by makiiig subdivisions of ABCvr/
straight lines through vertices (as in Prop, iil above).

Taking two equal triangles of this kind and placing their hypotennies
together, we obtain a quadrilateral with four right angles and four equal
sides.

Putting four of these quadrilaterals together, we obtain a new quadrilateral
of the same kind but with its sides double of those of the first quadrilateral

After n such operations we have a quadrilateral with four rigt^t angles and
four equal sides, each being equal to s* times the side AB.

The diagonal of this qiuidrilateral divides it into two equal isosceles ri^t-
angled triangles in each of whicb tibe sum of the angles is equal to two ng^t
angles.

Consequently, from the existence of one triansle in which the sum of the
three angles is equal to two right angles it follows ttiat there exists an isoscetes
right-angled triangle widi sides greater than any assigned redDineal segment
and such that the sum of its three angles is also equal to two right ang^

V. If the sum of the three angies of one triangle is equal to twoo rigkt
angles^ the sum of the three angks of any other triauffe is also ofual to twoo
right angies.

It is enough to prove this for a rigki-anf/ed triangle, since any triangle can
be divided into two right-angled triangles.

Let ABC be any right-angled triangle.

If then the sum of the angles of any one
triangle is equal to two right angles, we can
construct (by the preceding Prop.) an isosceles
right-angled triangle with the same property and
with its perpendicular sides greater than those of
ABC

Let ABC be such a triangle, and let it be
applied to ABC^ as in the figure.

Applying then Prop. in. above, we deduce
first that the sum of the three angles of the
triangle ABC is equal to two right angles, and

next, for the same reason, that the sum of the three angles of the original
triangle ABC is equal to two right angles.

VI. If in any one triangle the sum of the three an^s is less than two
right angles, the sum of the three angjUs of any other triangle is also less than
two right angles.

This follows from the preceding theorem.

(It will be observed that the last two theorems are included among those
of Saccheri, which contain however in addition the corresponding theorem
touching the case where the sum of the angles is greater than two right
angles.)

We come now to the bearing of these propositions upon Euclid's Postulate
5 ; and the next theorem is

VII. If the sum of the three angles of a trian^e is equal to two ri^
angles^ through any point in a plane there can only be drawn one parallel to a
givem straight line.

I

I. Post. 5] NOTE ON POSTULATE 5 319

For the proof of this we require the following

Lemma. // is a/ways possible^ through a point P, to draw a straight line
which shall mahe^ with a gh^en straight line (r), an angle less than any assigned
angle.

Let Q be the foot of the perpendicular from i'upon r.

Let a segment QR be taken on r,

on either side of Q, such that QR is
equal to PQ,

Join PR^ and mark off the segment
RR' equal to PR ; join PR\

If m represents the angle QPR or
the angle QRP^ each of the equal
angles RPR\ RR'P is not greater
than a>/2.

Continuing the construction, we obtain, after the requisite number of
operations, a triangle /^Rn-i Rn in which each of the equal angles is equal to
or less than m/a*.

Hence we shall arrive at a straight line PR,^ which, starting from /'and
meeting r, makes with r an angle as small as we please.

To return now to the Proposition. Draw from P the straight line s
perpendicular to PQ.

Then any straight line drawn from P which meets r in ^ will form equal
angles with r and x, since, by hypothesis, the sum of the angles of the triangle
PqR is equal to two right angles.

And since, by the I^mma, it is always possible to draw through i'strai^t
lines which form with r angles as small as we please, it follows that all the
straight lines through P^ except x, will meet r. Hence s is the only parallel
to r that can be drawn through P.

The history of the attempts to prove Postulate 5 or something equivalent
has now been brought down to the parting of the ways. The further
developments on lines independent of the Postulate, bc^ning with
Schweikart (1780 — 1857), Taurinus (1794 — 1874)^ Gauss (1777 — 1855),
Lobachewsky (1793 — 1856), J. Bolyai (1802— 1860) , Riemann (1826— 1866),
belong to the history of non-Euclidean geomefry, which is outside the scope
of this work. I may refer the reader to the full article Snlla teoria delie
parallele e suite geometrie non-euclidee by R. Bonola in Questioni riguardanti
I la geometria elementare^ 1900, of which I have made considerable use in the

[' above, to the same author's La geometria non-eudidea^ Bologna, 1906, to the
first volume of Killing's Einfiihrung in die Grundlagen der Geometrie^
Paderbom, 1893, to P. Mansion's Premiers principes de nUtaghmktrie^ and
) P. Barbann's La gtomttrie non-Euclidienne^ Paris, 1902, to the historical
' summary in Veronese's Fondamenti di geometria^ 1891, p. 565 sqq., and (for
original sourc^ to Engel and Stackel, Die Theorie der Parallellinien von
Euhlid bis auf Gauss^ 1895, ^^^ Urkunden %ur Geschichie der nicht-Euhlidischen
Geometriey i. (Lobachewsky), 1899, and 11. (Wolfgang und Johann Bolyai).
I will only add that it was Gauss who first expressed a conviction that the
Postulate could never be proved ; this he stated distinctly, first in a review in
the Gottingische gdehrte Anseigen, 20 A pril 1816, and secondly in a letter to
Bessel of 27 Januaiy, 1829. The actual inaemonstrability of the Postulate was
proved by Beltrami (1868) and by Hoiiel {Note sur Timpossibiliti de dSmontrtr
par une construetion plane le prindpe de la thkorie des paralUles dit Postulatum
d*Euclide in Battaglini's Giomale di matematiche^ viii., 1870, pp. 84 — 89).

»o BOOK I [vTon.s

Alternatives for Postulate 5.

It may be convenient to odiect here a few of the more ndewortby
substitutes which have from time to time been fbnnally suggested or tac^
assumed.

(i) Through a ghen paini mfy 9m parallel €an he drawn io a given
straight line or, Two straight Knes which interseet one another eamnat hath he
parallel to one and the same straight Hne.

This is commonly known as '' Playfair's Axiom," but it was of coorie not
a new discovery. It is distinctly stateo in Produs* note to EucL i. 31.

(i a) If a straight line interseei one rf twoo parallels^ it wHl inieruet the
other also (Proclus).

(i b) Straight lines parallel to the same straight line are paralM to one
another.

The forms (i a) and (i b) are exactly equivalent to (i).

(s) There exist straight Unes everywhirt oguidistemt firmn on€ emctker
(Posidonius and Geminus); with which may be compared Produs* ticit
assumption that Parallels remain^ throngkout their lenf^n^ at a finite distama
from one another.

(3) There exists a trian^e in whieh the sum of the three an^ iseqmUto
two right angles (L^endre).

(4^ Given any figure^ there exists a figure similar to it of any siu wepleau
(Walhs, Cftmot, Laplace).

Saccheri points out that it is not necessary to assume so much, and that it
is enough to postulate that there exist two unequal triemj^ with eqmd angles.

(5) Through any point within an angle less than two4hirds of a right at^
a straight line can always be drawn which meets both sides of the anj^e
(Legendre).

With this may be compared the similar axiom of Lorenz {Grundriss der
reinen und angewandten Mathematih^ 1791): Every straight line through a
point within an angle must meet one of the sides of the angle,

(6) Given any three points not in a straight line^ there exists a circle pcusing
through them (Le^ndre, W. Bolyai).

(7) . ^* If I could prove that a rectilineal triangle is possible the content of
which is greater than any given area^ I cm in a position to prove perfectly
rigorousfy the whole of geometry^ (Gauss, in a letter to W. Bolyai, 1799).

Cf. the proposition of Legendre numbered iv. above, and the axiom of
Worpitzky: TTiere exists no tricmgle in which every angU is cu small as we
please.

(8) If in a quadrilateral three angles are right angles^ the fourth angle is
a right angle also (Clairaut, 1741).

(9) If two straight lines are parcUld^ they are figures opposite to (or the
reflex of) one another with respect to the middle points of all their transversal
segments (Veronese, Elementi^ 1904).

Or, Two parallel straight lines intercept, on every transversal which passes
through the middle point of a segment included betwun them^ another segment
the middle point of which is the middle point of the first (Ingrami, ElemenH,
>904).

Veronese and Ingrami deduce immediately Playfair's Axiom,

;i

NOTES ON THE COMMON NOTIONS »ii

AXIOMS OR COMMON NOTIONS.

\

' In a paper Sur Pauthentkiti des axiomes ttEuclide in the Bulletin des

sciences nuUMmatiques et ustronamiques^ a* s^r. viii., 1884, p. 162 sqq., Paul

' Tannery maintained that the Common Notions (including the first three) were
not in Euclid's work but were interpolated later. The following are his main
aiguments. (i) If Euclid had set about distin^ishing between indemon-
stmble principles (a) common to all demonstrative sciences and (b) peculiar
to geometry, he would, says Tannery, certainly not have placed the common
principles second and the special principles (the Postulates) first. (2) If the
Common Notions are Euclid's, thb designation of them must be his too ; for he
must have used some name to distinguish them from the Postulates and, if he
had used another name, such as Axioms^ it is impossible to imagine why that
name was changed afterwards for a less suitable one. The word hn^wa
{notion\ ^ys Tannery, never signified a notion in the sense of a proposition^
but a notion of some odjat ; nor is it found in any technical sense in Plato
and Aristotle. (3) Tannery's own view was that the formulation of the
Common Notions dates from the time of Apollonius, and that it was inspired
by his work relating to the Elements (we know from Proclus that Apollonius
tried to prove the Common Notions), This idea, Tannery thought, was

J confirmed by a " fortunate coincidence " furnished by the occurrence of the
word Hyyota (notion) in a quotation by Proclus (p. 100, 6): ''we shall agree
with Apollonius when he says that we have a nolion (lirocay) of a line when
we order the lengths, only, of roads or walls to be measured."

In reply to argument (i) that it is an unnatural order to place the purely
geometrical Postulates first, and the Common Notions^ which are not peculiar
to geometry, last, it may be pointed out that it would surely have been a still
more awkward arrangement to give the Definitions first and then to separate
from them, by the interposition of the Common Notions^ the Postulates, which
are so dosely connected with the Definitions in that they proceed to postulate
the existence of certain of the things defined, namely straight lines and circles.

(2) Though it is true that wvow, in Plato and Aristotle is generally a
notion of an object^ not of difact or proposition, there are instances m Aristotle
where it does mean a notion of a fiact : thus in the Eth, Nic, ix. 11, ii7i a32
he speaks of ''the notion (or consciousness) that friends sympathise^ (17 &voia
rw (TvraXyciy rovs ^cXovs) and again, b 14, of "the notion (or consciousness)
that they are pleased at his good fortune." It is true that Plato and Aristotle
do not use the word in a technical sense ; but neither was there apparently in
Aristotle's time any fixed technical term for what we call "axioms," since he
speaks of them variously as " the so-called axioms in mathematics," " the so-
odled common axioms," "the common (things)" (ra xocm), and even "the
common opinions " (jcoiml 80^1). I see therefore no reason why Euclid should
not himself have given a technical sense to " Common Notions," which is at

I ' least a distinct improvement upon "common opinions."

(3) The use of hn^wa in Proclus' quotation from Apollonius seems to me
.. to be an unfortunate, rather than a fortunate, coincidence from Tannery's point

of view, for it is there used precisely in the old sense of the notion of an
object (in that case a line).

No doubt it is difficult to feel certain that Euclid did himself use the term
Common Notions^ seeing that Proclus' commentary generally speaks of Axioms.
But even Proclus (p. 194, 8), after explaining the meanmg of the word
"axiom," first as used by the Stoics, and secon<fiy as used by "Aristotle and

t32 • BOOK I [lCMi

the geometers," goes on to say : ''For in their view (that of Aristotk and the
geometers) axiom and common naium are the same thing." This, as it seems
to me, may be a sort of apology for using the word ''axiom " exdusivdy in
what has ^one before, as if Prochis had suddenly bethou^t himself that he
had descnbed both Aristotle and the geometers as using the one tenn
"axiom," whereas he should have said that Aristotle spoke of "axiomsi* while
"the geometers" (in fact EuclidX though meaning the same thing, odled them
Common Notions. It may be for a like reason that in another passage (p. 76,
16X after quoting Aristotle's view of an "axiom," as distinct from a portukfr
and a hypothesis, he proceeds : " For it is not by virtue of a iommom noHon
that, without being taught, we preconceive the circle to be such and such a
figure." If this view of the two passages just (quoted is correct, it would
strengthen rather than weaken the case for the genwneness of Common NoHoms
as the Euclidean term.

Again, it is clear from Aristotle's allusions to the "common axioms in
mathematics " that more than one axiom of this kind had a place in Uie text-
books of his day ; and as he constantly quotes the particular axiom that, y
eqnais be taken from equals^ the remainders are equals which is Eudid*^ Common
Notion ^ it would seem that at least die first three Common Notions were
adopted by Euclid from earlier text-books. It is, besides, scarcely credifa3e
that, if the Common Notions which ApoUonius tried to Drove had not been
introduced earlier (e.g. by Euclid), they would then have been interpolatad as
axioms and not as propositions to be proved. The line taken by Apollonius
is much better explained on the assumption that he was directly attacking
axioms which he found already admitted mto the Elements.

Proclus, who recognised die five Common Notions given in the text, warns
us, not only against the error of unnecessarily multiplying the axioms, but
against the contrary error of reducing their number unduly (p. 196, 15), "as
Heron does in enunciating three only; for it is also an axiom that the whole is
greater than the party and mdeed the geometer employs this in many places for
his demonstrations, and again that things which coinade are egucU**

Thus Heron recognised the first three of the Common Notions ; and this
foct, together with Aristotle's allusions to "common axioms*' (in the plural),
and in particular to our Common Notion 3, may satisfy us that at least the first
three Common Notions were contained in the Elements as they left Euclid's
hands.

Common Notion i.

Ta vf a^rf ura jcal o^Xi/Xotf joriy Zero.

Things which are equal to the same thing are also equal to one another,

Aristotle throughout emphasises the fact that axioms are self-evident truths,
which it is impossible to demonstrate. If, he says, any one should attempt to
prove them, it could only be through ignorance. Aristotle therefore would
undoubtedly have agreed in Proclus' strictures on Apollonius for attempting
to prove the axioms. Proclus gives (p. 194, 25), as a specimen
of these attempted proofs by Apollonius, that of the first of the
Common Notions. " Let A be equal to Bj and the latter to C;
I say that A is also equal to C. For, since A is equal to £/\t A B
occupies the same space with it ; and since B is equal to C, it
occupies the same space with it.

Therefore A also occupies the same space with C"

Proclus rightly remarkis (p. 194, 22) that "the middle term is no more

.
►

I. CM 1—3] NOTES ON COMMON NOTIONS 1—3 iti

intelligible (better known, yvwptfuirtfiov) than the conclusion, if it is not
actually more disputable." Again (p. 195, 6), the proof assumes two things,
(i) that things which "occupy the same space" (roiros) are equal to one
another, and (2) that things which occupy the same space with one and the
same thing occupy the same space with one another ; which is to explain the
obvious by something much more obscure, for space is an entity more
unknown to us than the things which exist in space.

Aristotle would also have objected to the proof that it is partial and not
general (xotfoXov), since it refers only to things which can be supposed to
occupy a space (or take up room), whereas the axiom is, as Proclus says
(p. 196, i), true of numbers, speeds, and periods of time as well, though of
course each science uses axioms in relation to its own subject-matter only.

Common Notions 2, 3.

2. Kai iav urocc icra irpo(rr€0j, ra oXa iartv urcu

3. Koi iav airo icrwy ccra dflHup€6i, ^^ KaroXciird/Acva ifrriv icro.

2. If equals be added to equals^ the wholes are equal,

3. If equals be subtracted from equals^ the remainders are equal.

These two Common Notions are recognised by Heron and Proclus as
genuine. The latter is the axiom which is so favourite an illustration with
Aristotle.

Following them in the mss. and editions there came four others of the
same type as i — ^^3. Three of these are given by Heiberg in brackets ; the
fourth he omits altogether.

The three are :

(a) If equals be adtled to unequals^ the wholes are unequal,

iff) Things which are double of the same thing are equal to one another,

(c) Things which are halves of the same thing are equal to one anotlur.
The fourth, which was placed between {a) and (^), was :

(d) If equals be subtracted from unequals^ the remainders are unequal,

Proclus, in observing that axioms ought not to be multiplied, indicates
that all should be rejected which follow ^m the five admitted by him and
appearing in the text above (p. 155). He mentions the second of those just
quoted {p) as one of those to be excluded, since it follows from Common
Notion I. Proclus does not mention (a), (r) or (d)\ an-NairizI gives (a), {fl\ (b)
and (^), in that order, as Euclid's, adding a note of Simplicius that '' three
axioms (sententiae acceptae) only are extant in the ancient manuscripts, but
the number was increased in the more recent"

(a) stands self-condemned because " unequal " tells us nothing. It is easy
to see what is wanted if we refer to i. 17, where the same angle is added to a
greater and a Uss^ and it is inferred that the first sum is greater than the second.
So far however as the wording of (a) is concerned, the addition of equal to
greater and less might be supposed to produce less and greater respectively. If
therefore such an axiom were given at all, it should be divided into two.
Heiberg conjectures that this axiom may have been taken from the commentary
of Pappus, who had the axiom about equals added to unequals quoted below
{ey^ if so, it can only be an unskilful adaptation of some remark of Pappus, for
his axiom {e) has some point, whereas (a) is useless.

As regards (b\ I agree with Tannery in seeing no sufficient reason why, if

t

J

i

224 BOOK I [l C.JK

we reject it (as we certainly must), the words in i. 47 '^But things irtiich aie
double of equals are equal to one another" should be condemned as an
interpolation. If they were interpoUted, we should have eipected to find the
same interpolation in i. 42, where the axiom is ioMy assumed. I diink
it quite possible that Euclid may have inserted such words in one case and
left them out in another, without necessarily implying either that he was
quoting a formal Common Naium of his own or that he had mi inchided
among his Common Notions the particular fiu:t stated as obvious.

The corresponding axiom (r) about the kahes of a^uals can hardly be
genuine if {b) is not, and Produs does not mention it Tannery acutdy
observes however that, when Heibeig, in i. 37, 38, brsckets words statiiw that
^'the halves of equal things are equal to one another** on the ground that
axiom {f) was interpolated (althou^ before Theon's time), and explains that
Euclid used Common Notion ^ in making his inference, he is clearly mistaken.
For, while axiom (b) is an obvious inference from Common Notion s, axiom lA
is not an inference from Common Notion 3. Tannery says, in a note, that (q
would have to be established by redtutio ad atsunhtm with the help of axiom
{b\ that is to say, of Common Notion s. But, as the hypothesis in ttie roAuHo
adabsurdum would be that one of the halves ispraier than the other, and it
would therefore be necessary to prove that the one whole is gnoUr than the
other, while axiom {b) or Common Notion % only refers to oqfuds^ a little
argument would be necessary in addition to the refimnce to Common Notion s.
I diink Euclid would not have gone through this process in order to prove (^),
but would have assumed it as equally obvious with (by

Proclus (pp. 197, 6— 19S, 5) definitely rejects two other axioms of the
above kind given by Pappus, observing that, as they follow fimn the {pmine
axioms, they are rightly omitted in most copies, although Pappus said that
they were '' on record " with the others (w¥aj¥aypau^wBai) :

(e) If unequais be added to equals^ the difference between the wholes is equal
to the difference between the added parts \ and

(/) If equals be added to unequais^ the difference between the wholes is equal
to the difference between the original unequcUs.

Proclus and Simplidus (in an-Nairi2l) give proofs of both. The proof of
the former, as given by Simplicius, is as follows :

Let A£, CD be equal magnitudes ; and let E£, FD be £
added to them respectively, EB being greater than FD. q

I say that AE exceeds CF by the same difference as that by
which BE exceeds DF B

Cut ofif from BE the magnitude BG equal to DF.

Then, since AE exceeds AG by GE^ and AG is equal to CF
and BG to DF,

AE exceeds CF by the same difference as that by which BE
exceeds DF. I

Common Notion 4. ^

Kal ra i^apfi6(,oyra cir* oXXi^Aa ura aXXi/Xoif coTiK. *

7%ings whkh coincide with one another are equal to one another.
The word c^kpfuSfctv, as a geometrical term, has a different meaning |
according as it is used in the active or in the passive. In the passive, |
i^apfiHw^ai, it means "to be applied to" without any implication that the
app^ed figure will exactly fit, or coincide with, the figure to which it is applied;
on the other hand the active i^apfioC€tv is used intransitively and means ''to

T

r

I. C N. 4] NOTES ON COMMON NOTIONS 2—4 ^^5

fit exactly," "to coincide with." In Euclid and Archimedes i^apyuUgmif is
constructed with ciri and the accusative, in Pappus with the dative.

On Common Notion 4 Tannery observes that it is incontestably geometrical
in character, and should therefore have been excluded from the Common
Notions; again, it is difficult to see why it is not accompanied by its converse,
at all events for straight lines (and, it might be added, angles also), wluch
Euclid makes use of in i. 4. As it is, says Tannery, we have here a definition
of geometrical equality more or less sufficient, but not a real axiom.

It is true that Proclus seems to recognise this Common Notion and the next
as proper axioms in the passage (p. 196, 15 — 21) where he says that we should
not cut down the axioms to the minimum, as Heron does in giving only three
axioms; but the statement seems to rest, not upon authority, but upoo an
assumption that Euclid would state explicitly at the b^inning all axioms
subsequently used and not reducible to others unquestionably included. Now
in I. 4 this Common Notion is not quoted ; it is simply inferred that " the base
BC will coincide with EF^ and will be equal to it." The position is therefore
the same as it is in regard to the statement in the same propositipn that, *Hf...
the base BC does not coincide with EF^ two straight lines will enclose a sface :
which is impossible " ; and, if we do not admit that Euclid had the axiom tiiat
" two straight lines cannot enclose a space," neither need we infer that he had
Common Notion 4. I am therefore inclined to think that the latter is mcn-e
likely than not to be an interpolation.

It seems clear that the Common Notion, as here formulated, is intended
to assert that superposition is a legitimate way of proving the equality of two
figures which have the necessary parts respectively equal, or, in other wqrIs,
to serve as an axiom of congruence.

The phraseology of the propositions, e.g. i. 4 and i. 8, in which Eodid
employs the method indicated, leaves no room for doubt that he regarded one
figure as actually moved and placed upon the other. Thus in i. 4 he ays,
"The triangle ABC being applied (c^op/tofo/Acvov) to the triangle DEF^wcA
the point A being plaad {jtM^ivwi) upon the point />, and the straight line
AB on DE^ the point B will also coincide with E because AB is equd to
DE''\ and in i. 8, " If the sides BA^ AC do not coincide with ED, DF, but
fall beside them (take a different position, iropaXXa^ouo-iF), then " etc At the
same time, it is clear that Euclid disliked the method and avoided it whereirer
he could, e.g. in i. 26, where he proves the equality of two triangles which have
two angles respectively equal to two angles and one side of the one equd to
the corresponding side of the other. It looks as though he found the method
handed down by tradition (we can hardly suppose that, if Thales proved tiiat

) the diameter of a circle divides it into two equal parts, he would do so by any
other method than that of superposition), and followed it, in the few cues
where he does so, only because he had not been able to see his way to a

i satisfactory substitute. But seeing how much of the Elements depends on l 4,
directly or indirectly, the method can hardly be regarded as being, in Eodid,
of only subordinate importance ; on the contrary, it is fundamental Nor, as

< a matter of fact, do we find in the ancient geometers any expression of doubt
as to the legitimacy of the method. Archimedes uses it to prove that any

. spheroidal figure cut by a plane through the centre is divided into two equal

I parts in respect of both its surface and its volume; he also postulates in
Equilibrium of Planes i. that " when equal and similar plane figures coincide
if applied to one another, their centres of gravity coincide also."

j Killing (Einfuhrung in die Grundlagen der Geometrie^ 11. pp. 4, 5)

H. E. 15

236 BOOK I [lCMa

contrasts the attitude of the Greek geometers with that of the philosc^riieni
who, he says, appear to have agreed in banishing motion from geometry
altogether. In support of this he refers to the view frequently expressed l^
Aristotle that mathematics has to do with immovable objects (aiciri|raX and that
only where astronomy is admitted as part of mathematical science is motion
mentioned as a subject for mathematics. Cf. MiU^h. 989 b 32 ''For madie-
matical objects are among things which exist i^Mut from motion, except sudi
as relate to astronomy"; Metapk. 1064 a 30 ''Physics deals with things
which have in themselves the principle of motion; mathematics is a theoretical
science and one concerned with things which are siatumary (/imirra) but not
separable" (sc. from matter^; in Physics 11. 2, 193 b 34 he speaks of the
subjects of mathematics as "m thought separable from motion."

But I doubt whether in Aristotle's use of the words "immovaUe," "widi-
out motion" etc as applied to the subjects of mathematics diere is any
implication such as Killing supposes. We arrive at mathematical concepts
by abstraction from material objects ; and just as we, in thought, eliminate
the matter, so according to Aristotle we diminate the attributes of matter as
such, e.g. qualitative clmnge and motion. It does not appear jto me that die
use of " immovable " in the passages referred to means more than this. I do
not think that Aristotle would have r^purded it as illegitimate to mom z.
geometrical figure from one position to another; and I infer this from a
passage in De caelo iii. i where he is criticising "those who make up every
iKxly that has an origin by putting together pianes^ and resolve it again into
planesJ' The reference must be to the TimaeHS (54 B sqq.) where Plato
evolves the four elements in this way. He begins with a right-angled triangle
in which the hypotenuse is double of the smaller side; six of these put together
in the proper way produce one equilateral triangle: Making solid uigles with
(a) three, (b) four, and (c) five of these equilat^al triangles respectively, and
taking the requisite number of these solid angles, namely four of (tf), six of (b)
and twelve of (c) respectively, and putting them together so as to form regular
solids, he obtains (a) a tetrahedron, (fi) an octahedron, (y) an icosahedron
respectively. For the fourth element (earth), four isosceles right-angled triangles
are first put together so as to form a square, and then six of these squares are
put together to form a cube. Now, says Aristotle (299 b 23), '*it is absurd that
planes should only admit of being put together so as to touch in a line', for just
as a line and a line are put together in both ways, lengthwise and breadthwise, r
so must a plane and a plane. A line can be combined with a line in the sense
of being a line superposed^ and not adde(P^\ the inference being that deplane can |
be superposed on biplane. Now this is precisely the sort of motion in question !
here; and Aristotle, so far from denying its permissibility, seems to blame ^
Plato for not using it. Cf. also Physics v. 4, 228 b 25, where Aristotle speaks '
of *' the spiral or other magnitude in which any part will not coincide with ,
any other part," and where superposition is obviously contemplated.

Motion without deformation.

It is well known that Helmholtz maintained that geometry requires us to
assume the actual existence of rigid bodies and their free mobility in space, .
whence he inferred that geometry is dependent on mechanics.

Veronese exposed the fallacy in this (Fondamenti di geometria^ pp. xxxv —
^Dovi, 239 — 240 note, 615 — 7), his argument being as follows. . Since geometry
is concerned with empty space, which is immovable, it would be at least strange
if it was necessary to have recourse to the real motion of bodies for a definition.

i

I UC.M4] NOTE ON COMMON NOTION 4 227

and for the proof of the properties, of immovable space. We must distinguish
\ the intuitive principle of motion in itself from that of motion without deforma-
tion. Every point of a figure which moves is transferred to another point in
space. " Without deformation " means that the mutual relations between the
points of the figure do not change, but the relations between them and other
figures do change (for if they did not, the figure could not move). Now
consider what we mean by saying that, when the figure A has moved from
the position Ax to the position ^,, the relations between the points of A in
the position A^ are unaltered from what they were in the position A^^ are the
same in fact as if ^ had not moved but remained at A^^ We can only say
that, judging of the figure (or the body with its physical qualities eliminated)
by the impressions it produces in us during its movement, the impressions
produced in us in the two different positions (which are in tim^ distinct)
are equal. In fact, we are making use of the notion of equality between two
distinct figures. Thus, if we say that two bodies are equal when they
can be superposed by means of movement without deformation^ we are com-
mitting di petitio principii. The notion of the equality of spaces is really prior
to that of rigid bodies or of motion without deformatioa Helmholtz supported
his view by reference to the process of measurement in which the measure
must be, at least approximately, a rigid body, but the exbtence of a rigid body
as a standard to measure by, and the question how we discover two equal
spaces to be equal, are matters of no concern to the geometer. The method
of superposition, depending on motion without deformation, is only of use as
9i practical test \ it has nothing to do with the theory of geometry.

Compare an acute observation of Schopenhauer (Die Welt als WilU^ 2 ed.
1844^ II. p. 130) which was a criticism in advance of Helmholtz' theory : "I
am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the
eighth is not rather attacked : ' Figures which coincide (sich decken) are
equal to one another.' For coincidence (das Sichdecken) is either mere
tautology, or something entirely empirical, which belongs, not to pure intuition
(Anschauung), but to external sensuous experience. It presupposes in fact
the mobility of figures; but that which is movable in space is matter and
nothing else. Thus this appeal to coincidence means leaving pure space, the
sole element of geometry, in order to pass over to the material and empirical."

Mr Bertrand Russell observes (En^clopaedia BritannicOy Suppl. Vol. 4,
1902, Art '* Geometry, non-Euclidean ") that the apparent use of motion here
is deceptive ; what in geometry is called a motion is merely the transference
of our attention from one figure to another. Actual superposition, which is
nominally employed by Euclid, is not required; all that is required is the
transference of our attention from the original figure to a new one defined by
the position of some of its elements and by certain properties which it shares
with the original figure.

If the method of superposition is given up as a means of defining theoreti-
cally the equab'ty of two figures, some other definition of equality is necessary.
But such a definition can be evolved out of empirical or practical observation
of the result of superposing two material representations of figures. This is
done by Veronese (Ekmenti di geometria^ 1904) and Ingrami (Elementi di
geometria, 1904). Ingrami says, namely (p. 66):

" If a sheet of paper be folded double, and a triangle be drawn upon it
and then cut out, we obtain two triangles superposed which we in practice call
equal. If points ^, J?, C, Z> ... be marked on one of the triangles, then,
when we place this triangle upon the other (so as to coincide with it), we see

IS— 2

228 BOOK I [l C. N. 4

that each of the particular points taken on the first is superposed on one
particular point of the second in such a way that the segments AB^ AC, AD,
BQ BD^ CDy ... are respectively superix>sed on as many segments in die
second triangle and are therefore equal to them respectively. In this way we
justify the following

'' Definition of equality.

''Any two figures whatever will be called equal when to the pdnts of one
the points of the other can be made to correspond univacalh [Le. every 0me
point in one to one distinct point in the other and viu vena] in such a way
that the segments which join the points, two and two, in one figure are
respectively equal to the segments which join, two and two, the corresponding
pomts in the other."

Ingrami has of course previously postulated as known the signification of
the phrase equa/ {rectilineal) segments, of which we get 9i practical notion when
we can place one upon the other or can place a third movable segment
successively on both.

New systems of Congruence-Postulates.

In the third Article of Questiom riguardanti la geametria ekmentart, ipop,
pp. 65 — 82, a review is given of three difierent systems: U) that of Pas(£ in
Vorl^ngen Oder neuere Geamdrk, 1882, p. loi sqq., (2) that of Veronese
according to the FondammH H geametria, 1891, md die ElemenH taken
together, (3) that of Hilbert (see Chundlagen der Geametrie, 1903, pp. 7 — 15).

These systems difier in the particular conceptions taken by the three
authors as primary, (i) Pasch considers as primary the notion of congruence
or equality between any figures which are made up of a finite number of points
only. The definitions of congruent segments and of congruent angles have to
be deduced in the way shown on pp. 68 — 9 of the Article refened to, after
which EucL i. 4 follows immediately, and Eucl. i. 26 (i) and i. 8 by a
method recalling that in Eucl. i. 7, 8.

(2) Veronese takes as primary the conception of congruence between
segments (rectilineal). The transition to congruent angles^ and thence to
triangles is made by means of the following postulate :

''Let AB^ AC and A'B\ AC be two pairs of straight lines intersecting
at A^ A\ and let there be determined upon them the congruent segments
AB, AB* and the congruent segments AC, AC\

then, if BC^ B*C are congruent, the two pairs of straight Una are con-
gruent"

(3) Hilbert takes as primary the notions of congruence between twth
segments and angles, .

It is observed in the Article referred to that, from the theoretical stand- [
point, Veronese's system is an advance upon that of Pasch, since the idea of
congruence between segments is more simple than that of congruence between t
any figures ; but, didactically, the development of the theory is more compli- '
cated when we start from Veronese's system than when we start from that of \
Pasch.

The system of Hilbert offers advantages over both the others from the
point of view of the teaching of geometry, and I shall therefore give a short
account of his system only, following the Article above quoted.

i

I. C, N. 4] NOTE ON COMMON NOTION 4 229

Hubert's system.

The following are substantially the Postulates laid down.

(i) If one s^mmt is congruent ivith another^ the second is also congruent
with the first,

(2) If an angle is congruent with another angU^ the second angle is also
congruent with the first,

(3) Two segments congruent with a third are congruent with one another,

(4) 7\vo angles congruent with a third are congruent with one another,

(5) Any segment AB is congruent with itself^ independently of its sense.
This we may express symbolically thus :

AB^AB^BA,

(6) Any angle (ab) is congruent with itself independently of its sense.
This we may express symbolically thus :

(ab) = {ab) = (ba),

(7) On any straight line r*, starting from any one of its points A', and on
each side of it respectively^ there exists one and only one segment congruent with a
segmeftt AB belonging to the straight line r.

(8) Given a ray a, issuing from a point O, in any plane which contains it
and on each of the two sides of it^ there exists one and only one ray b issuing
from O stich that the angle (ab) is congruent with a given angle (a'b').

(9) ^ AB, BC are two consecutive segments of the same straight line r
{segments^ that is, having an extremity and no other point common), and A'B',
B'C two consecutive segments on another straight line r', and if AB = A'B',
BC s B'C, then

AC^AC,

I (10) j^ (ab), (be) are two consecutiife angles in the same plane ir (angles,

(that is, having the ifertex and one side common), and (a'b'), (b'c') two consecu-
tive angles in another plane v, and if (A) = (a'b'), (be) = (b'c'), then

^ (11) If two trian^s have two sides and the included angles respectively

congruent, they have also their third sides congruent as well as the angles
opposite to the congruent sides respectiwly,

I As a matter of fact, Hilbert's postulate corresponding to ^11) does not

(

assert the equality of the third sides in each, but only the equality of the two
remaining angles in one triangle to the two remaining angles in the other
respectively. He proves the equality of the third sides (thereby completing
the theorem of Eucl. i. 4) by reductio

ad absurdum thus. ljtiABC,AB'C' A A'

be the two triangles which have the ^^ ^y^^

sides AB, AC respectively congruent ^/^ \ ^/^ A

with the sides AB', A'C and the ^^ \ y^ \ \

included angle at A congruent with ^ ^ ^7- g-jj.

the included anple at A\

Then, by Hilbert's own postulate, the angles ABC, ABC are congruent,
as also the angles ACB, AC'B,

If BC is not congruent with BC, let D be taken on BC such that BC,
BD are congruent, and join AD,

7

930 BOOK I [i. C.N.A

Then the two triangles ABQ ASD have two rides and the included
angles congraent respectively; therefore, by the same postulate^ the angles
BACy BAD are congruent

But the angles BAC^ BA'C are congruent ; therefore, by (4) above, the
angles BAC^ BAD are congruent : which is impossible, since it contradicts
(8) above.

Hence BC^ BC cannot but be congruent

Eucl. I. 4 is thus proved \ but it seems to be as well to indude all of that
theorem in the postulate, as b done in ^i i) above, rince the two parts of it are
equally suggested by empirical observation of the result of one superporition.

A proof similar to that just given immediately establishes Eud. i. 26 (i),
and Hilbert next proves that

If two angles ABC, A'B'C are eangnteni with one another^ their supplt-
mentary angles CBD, CB'D' are also eo^pnent with one another.

We choose Ay D on one of the straight lines forming the first angle, and
A\ D on one of those forming the second angle, and agam C, C on the other

O C

/^^^^^

straight lines forming the angles, so that AB is congruent with AB^ C*B
with CB, and DB with DB.

The triangles ABC, ABC are congruent, by (11) above; and AC is
congruent with AC, and the angle CAB with the angle CAB.

Thus, AD, AD being congruent, by (9), the triangles CAD, CAB are
also congruent by (11); /

whence CD is congruent with CB, and the angle ADC with the angle
ADC.

Lastly, by (11), the triangles CDB, CDB are congruent, and the angles
CBD, CBD are thus congruent

Hilbert's next proposition is that |

Given that the angle (h, k) in the plane a is congruent with the angle (h', k') I
in the plane a, and that 1 is a half-ray in the plane a starting from the vertex |
of the angle (h, k) and lying within that angle, there always exists a half-ray V [
in the second plane a, starting from the vertejC of the an^ (h', k') and lying i
within that an^ such that y

(h,i)=(h'.i'). ««/(k.i)=(k',i'). •

If O, O are the vertices, we choose points A, B on h, h, and points A, B t
on K, K respectively, such that OA, OA are congruent and also OB, OB: *

\

(

The triangles OAB, OAB are then congruent ; and, if / meets AB in C,
we can determine C on AB such that ^'C is congruent with AC.
Then f drawn from O through C is the half-ray required.

T"

I. C N. 4]

NOTE ON COMMON NOTION 4

a3i

The congruence of the angles (^ /), {h\ f) follows from (11) directly, and
that of (k, i) and (^, f) follows in the same way after we have inferred by
means of (9) that, AB, AC being respectively congruent with A'F, A*C\ the
difference BC is congruent with the difference ffC\

It is by means of the two propositions just given that Hilbert proves that
All right angles are congruent with one another.

Let the angle BAD be congruent with its adjacent angle CAD^ and
Ukewise the angle BAU congruent with its adjacent angle CA'Jff. All four
angles are then right angles.

O'T

QW

D'

If the angle BA'D is not congruent with the angle BAD, let the angle
with AB for one side and congruent with the angle BA*D be the angle
BAD\ so that AD' falls either within the angle BAD or within the angle
DAC. Suppose the former.

By the last proposition but one (about adjacent angles), the angles
BA D^ BAD* being congruent, the angles CA*D, CAD' are congruent

Hence, by the hypothesis and postulate (4) above, the angles BAD\
CAD* are also congruent

And, since the angles BAD, CAD are congruent, we can find within the
angle CAD a half-ray CAD'* such that the angles BAD\ CAD'' are
congruent, and likewise the angles DAD\ DAD" (by the last proposition).

But the angles BAD\ CAD' were congruent (see above); and it
follows, by (4), that the angles CAD'^ CAD" are congruent: which is
impossible, since it contradicts postulate (8).

Therefore etc

Euclid I. 5 follows directly by applying the postulate (11) above to ABC,
A CB as distinct triangles.

Postulates (9), (10) above give in substance the proposition that "the
sums or differences of s^;ments, or of
angles, respectively equal, are eqxiaL"

Lastly, Hilbert proves EucL i. 8 by
means of the theorem of Eud. i. 5 and
the proposition just stated as applied to
angles.

ABC, A'ffC being the given triangles
with three sides respectively congruent,
we suppose an angle CBA" to be deter-
mined, on the side of BC opposite to A,
congruent with the angle A'BC, and we make BA" equal to A'B,

The proof is obvious, being equivalent to the alternative proof often given
in our text-books for Eud. i. 8.

232 BOOK I [i. CN.S

Common Notion 5.

fcoi ri i\o¥ rw fiifHw^ fi€l(fi¥ [Ivrtr].
7^ whole is greater than the pari.

Proclus includes this "axiom " on the same ground as the preceding one.
I think however there is force in the objection which Tannery takes to it,
namely that it replaces a difertni expression in EudL i. 6, where it is stated
that "the triangle DBC wdl be equal to the triangle ACB^ the ks$ to the
greater: which is absurd!* The axiom appears to be an abstraction or
generalisation substituted for an immediate inference from a geometrical
figure, but it takes the form of a sort of definition of whole and part The
probabilities seem to be against its being genuine, notwithstanding Proclus*
approval of it

Qavius added the axiom that the whok is the equal to the sum of its parts.

Other Axioms introduced after Euclid's time.

[9] Ikuo straight lines do not enclose (or eoniain) a space.

Proclus (g. 196, 21) mentions this in illustration of the undue multiplication
of axioms, and he points out, as an objection to it, that it belongs to the
subject matter of geometry, whereas axioms are of a general character, and
not peculiar to any one science. The real objection to the axiom is that it is
unnecessary, since the fact which it states is included in the meanina; of
Postulate I. It was no doubt taken firom the passage in i. 4, ''if...the oase
BC does not coincide with EF^ two straight lines wul enclose a space-, which
is impossible** \ and we must certainly re^ud it as an interpolation, notwith-
standing that two of the best iiss. have it after Postulate 5, and one gives it
as Common Notion 9.

Pappus added some others which Proclus objects to (p. 198, 5) because
they are either anticipated in the definitions or follow firom them.

te) ' All the parts of a plane^ or of a straight line, coincide with one another.

(h) A point divides a line^ a line a surface, and a surf cue a solid \ on which
Proclus remarks that everything is divided by the same things as those by
which it is bounded.

An-Nairizi (ed. Besthom-Heibcrg, p. 31, ed. Curtze, p. 38) in his version
of this axiom, which he also attributes to Pappus, omits the reference to
solids, but mentions planes as a particular case of surfaces.

" (a^ A surface cuts a surface in a line ;
(P) If two surfaces which cut one another are plane, they cut one another

in a straight line ;
(y) A line cuts a line in a point (this last we need in the first proposition)." ^

{h) Magnitudes are susceptible of the infinite (or unlimited) both by way of
addition and by way of successive diminution, but in both cases potentially only [
M airttfioy iv rocs iirfiOtfriv iartv jcou rg irpo<rO€(r€i jccu rg iinKa$<up4a'€L, Swofici
Sk heirtpov). i

An-Nairizl's version of this refers to straight lines and plane surfaces only : '
"at regards the straight line and the plane surface, in consequence of their .
evenness, it is possible to produce them indefinitely.

This "axiom" of Pappus, as quoted by Proclus, seems to be taken directly I
firom the discussion of ro an-cipov in Aristotle, Physia in. 5 — 8, even to the
wording, for, while Aristotle uses the term division (8ia(Jpc(ric) most frequently
as the antithesis of addition (<rw0c(ricV he occasionally speaks of subtraction
(d^aipco-if) and diminution (Ka^oipco-ic). Hankel {Zur Geschichte der Mathe-
matih im Alterthum und MittelaUer, 1874, pp. 119 — 120) gave an admirable

f

\

1. Axx.] ADDITIONAL AXIOMS 233

summary of Aristotle's views on this subject ; and they are stated in greater
detail in Gorland, AristoteUs und die Mathematik^ Marburg, 1899, PP* ^57 —
183. The infinite or unlimited (dirc4f>ov) only exists potentially (8wa/ict), not
in actuality (^cpycti^). The infinite is so in virtue of its endlessly changing
into something else, like day or the Olympic Games {Phys, in. 6, 206 a 15 — 25).
The infinite is manifested in different forms in time, in Man, and in the
division of magnitudes. For, in general, the infinite consists in something new
being continually taken, that something being itself always finite but always
different. Therefore the infinite must not be regarded as a particular thing
(to3c ri), as man, house, but as being always in course of becoming or decay,
and, though finite at any moment, always different from moment to moment.
But there is the distinction between the forms above referred to that, whereas
in the case of magnitudes what is once taken remains, in the case of time and
Man it passes or is destroyed but the succession is unbroken. The case of
addition is in a sense the same as that of division ; in the finite magnitude the
former takes place in the converse way to the latter ; for, as we see the finite
magnitude divided ad infinitum^ so we shall find that addition gives a sum
tending to a definite limit I mean that, in the case of a finite magnitude,
you may take a definite fraction of it and add to it (continually) in the same
ratio ; if now the successive added terms do not include one and the same
magnitude whatever it is [i.e. if the successive terms diminish in geometrical
progression], you will not come to the end of the finite magnitude, but, if the
ratio is increased so that each term does include one and the same magnitude
whatever it is, you will come to the end of the finite magnitude, for every
finite magnitude is exhausted by continuaUy taking from it any definite
fraction whatever. Thus in no other sense does the infinite exist, but only
in the sense just mentioned, that is, potentially and by way of diminution
(206 a 25 — b 13). And in this sense you may have potentially infinite
addition^ the process being, as we say, in a manner, the same as with division
ad infinitum : for in the case of addition you will always be able to find some-
thing outside the total for the time being, but the total will never exceed every
definite (or assigned) magnitude in the way that, in the direction of division,
the result will pass every definite magnitude, that is, by becoming smaller
than it The infinite therefore cannot exist even potentially in the sense of
exceeding every finite magnitude as the result of successive addition (206 b
16 — 22). It follows that the correct view of the infinite is the opposite of
that commonly held : it is not that which has nothing outside it, but that
which always has something outside it (206 b 33 — 207 a i).

Contrasting the case of number and magnitude, Aristotle points out that
(i) in number there is a limit in the direction of smallness, namely unity, but
none in the other direction : a number may exceed any assigned number
however great ; but (2) with magnitude the contrary is the case : you can find
a magnitude smaller than any assigned magnitude, but in the other direction
there is no such thing as an infinite magnitude (207 b r-^5). The latter
assertion he justified by the following argument However large a thin^ can
be potentially, it can be as large actually. But there is no magnitude
perceptible to sense that is infinite. Therefore excess over every assigned
magmtude is an impossibility; otherwise there would be something lu^er
than the universe (o^yos) (207 b 17 — 21).

Aristotle is aware that it is essentially of physical magnitudes that he is
speaking. He had observed in an earlier passage {Phys, in. 5, 204 a 34) that
it is peiiiaps a more general inquiry that would be necessary to determine

I

234 BOOK I [i. Axx.

whether the infinite is possible in mathematics, and in the domain of thought
and of things which have no magnitude; but he excuses himself from entenng
upon this inquiry on the ground that his subject is physics and sensible
objects. He returns however to the bearing of his conclusions on mathematics
in III. 7, 207 b 27 : "my argument does not even rob mathematicians of their
study, although it denies the existence of the infinite in the sense of actual
existence as something increased to such an extent that it cannot be gone
through (dSic^tryrov) ; for, as it is, thc^ do not even need the infinite or use
it, but only require that the finite (straight line) shall be as long as they please;
and another magnitude of any size whatever can be cut in the same ratio as
the greatest magnitude. Hence it will make no difference to them for the
purpose of demonstration."

Lastly, if it should be urged that the infinite exists in thought^ Aristotle
replies that this does not involve its existence in fact, A thing is not greater
than a certain size because it is conceived to be so, but because it i>; and
magnitude is not infinite in virtue of increase in thought (208 a 16 — 22).

Hankel and Gorland do not quote the passage about an infinite series of
magnitudes (206 b 3 — 13) included in the above paraphrase; but I have
thought that mathematicians would be interested in the distinct e3q>ression of
Aristotle's view that the existence of an infinite series the terms of which are
magnitudes is impossible unless it is convergent, and (with reference to
Riemann's developments) in the statement that it does not matter to geometiy
if the straight line is not infinite in length, provided that it is as long as we
please.

Aristotle's denial of even the potential existence of a sum of magnitudes
which shall exceed every definite magnitude was, as he himself implies, in
conflict with the lemma or assumption used by Eudoxus (as we infer firom
Archimedes) to prove the theorem about the volume of a pyramid. The
lemma is thus stated by Archimedes (Quadrature of a parabola^ preface):
" The excess by which the greater of two unequal areas exceeds the less can,
if it be continually added to itself, be made to exceed any assigned finite
area." We can therefore well understand why, a century later, Archimedes
felt it necessary to justify his own use of the lemma as he does in the same
preface : " The earlier geometers too have used this lemma : for it is by its
help that they have proved that circles have to one another the duplicate
ratio of their diameters, that spheres have to one another the triplicate ratio
of their diameters, and so on. And, in the result, each of the said theorems
has been accepted no less than those proved without the aid of this lemma."

Principle of continuity.

The use of actual construction as a method of proving the existence of J
figures having certain properties is one of the characteristics of the Elements.
Now constructions are effected by means of straight lines and circles drawn I
in accordance with Postulates i — 3 ; the essence of them is that such straight ^
lines and circles determine by their intersections other points in addition to \
those given, and these points again are used to determine new lines, and so on.
This being so, the existence of such points of intersection must be postulated )
or proved in iht same way as that of the lines which determine them. Yet ]
there is no postulate of this character expressed in Euclid except Post. 5. /
This postulate asserts that two straight lines meet if they satisfy a certain
condition. The condition is of the nature of a &opiirfio« (discrimination^ or
condition of possibility) in a problem ; and, if the existence of the point of

r

I.AXX.] PRINCIPLE OF CONTINUITY 235

intersection were not granted, the solutions of problems in which the points of
intersection of straight lines are used would not in general furnish the required
proofs of the existence of the figures to be constructed.

But, equally with the intersections of straight lines, the intersections of
circle with straight line, and of circle with circle, are used in constructions.
Hence, in addition to Postulate 5, we require postulates asserting the actual
existence of points of intersection of circle with straight line and of circle
with circle. In the very first proposition the vertex of the required equilateral
triangle is determined as one of the intersections of two circles, and we need
therefore to be assured that the circles will intersect Euc lid seems to assume
U ita s obvious, although it is not so ; and he makes a similar assumption m
XTii, It is true that in the latter case Euclid adds to the enunciation that
two of the given straight lines must be together greater than the third ; but
there is nothing to show that, if this condition is satisfied, the construction is
always possible. In i. 12, in order to be sure that the circle with a given
centre will intersect a given straight line, Euclid makes the circle pass through
a point on the side of the line opposite to that where the centre is. It appears
therefore as if, in this case, he based his inference in some way upon the
, definition of a circle combined with the fact that the point within it called
the centre is on one side of the straight line and one point of the circumference
on the other, and, in the case of two intersecting circles, upon similar con-
siderations. But not even in Book in., where there are several propositions
about the relative positions of two circles, do we find any discussion of the
conditions under which two circles have two, one, or no point common.

The deficiency can only be made good by the Principle of Continuity,

Killing {Einfiihrung in die Grundi^gen der Geometrie^ 11. p. 43) gives the
following forms as sufficient for most purposes.

(a) Suppose a line belongs entirely to a figure which is divided into two
parts ; then, if the line has at least one point common with each part, it must
also meet the boundary between the parts; or

(p) If a point moves in a figure which is divided into two parts, and if it
belongs at the beginning of the motion to one part and at the end of the
motion to the other part, it must during the motion arrive at the boundary
between the two parts.

In the Questioni riguardanti la geometria elementare^ Article 4 (pp. 83 — loi),
the principle of continuity is discussed with special reference to the Postulate
of Dedekmd, and it is shown, first, how the Postulate may be led up to and,
secondly, how it may be applied for the purposes of elementary geometry.

Suppose that in a segment AB of a straight line a point C determines
two segments A C, CB, If we consider the point C as belonging to only one
of the two segments ACy CB^ we have a division of the segment AB into
two parts with the following properties.

1. Every point of the segment AB belongs to one of the two parts.

2. The point A belongs to one of the two parts (which we will call the
firsi) and the point B to the other ; the point C may belong indifferently to
one or the other of the two parts according as we choose to premise.

3. Every point of the first part precedes every point of the second in the
order AB of the segment

(For generality we may also suppose the case in which the point C fedls at
A or at B. Considering C, in these cases respectively, as belonging to the
first or the second part, we still have a division into parts which have the
properties above enunciated, one part being then a single point A or B.)

L.

236 BOOK I [lAxx.

Now, considering carefully the inverse of the above proposition, we see
that it agrees with the idea which we have of the continuity of the straight
line. Consequently we are induced to admit as 9,poshdaU the following.

If a segment of a straight line AB is divided into two parts so that
{iS every point of the segment AB bdongs to one of theparts^

(2) t?u extremity A belongs to the first part and "^ to the second^ and

(3) any point whatever of the first part precedes any point whatever ef ike
seamd party in the order AB of the segment^

there exists a point C of the segment AB (which may belong either to ame
part or to the other) such that every point of AB that precedes C belongs to the
first party and every point of AB that follows C belongs to the second part in
the division originally assumed

(If one of the two parts consists of the single point A or B^ the point C
is the said extremity A ox B oi the segment)

This is the Postulate of Dedekind, which was enunciated by Dedekind
himself in the following slightly different form (Stetigheit und irrationale ZcMen^
1872, new edition 1905, p. 11).

'' If all points of a straight line fall into two classes such that every paint of
the first class lies to the left of every point of the second class^ there exists one emd
only one point which produces this division of all the points into two classes^ this
division of the straight line into two parts,**

The above enunciation may be said to correspond to the intuitive notion
which we have that, if in a segment of a straight line two points start from
the ends and describe the segment in opposite senses, they meet in a point
The point of meeting might be regarded as belonging to both parts, but for
the present purpose we must regard it as belonging to one only and subtracted
from the other part

Application of DedehincPs postulate to angles.

If we consider an angle less than two right angles bounded by two rays
a, by and draw the straight line connecting Ay a point on a, with By a point 4
on by we see that all points on the finite s^ment AB correspond univocally to
all the rays of the angle, the point corresponding to any ray being the point
in which the ray cuts the segment AB \ and if a ray be supposed to move
about the vertex of the angle from the position a to the position by the
corresponding points of the segment AB are seen to follow in the same
order as the corresponding rays of the angle {ab).

Consequently, if the angle (ab) is divided into two parts so that

(i) each ray of the angle (ab) belongs to one of the two parts,

it) the outside ray a belongs to the first part and the ray b to the second,

(3) any ray whatever of the first part precedes any ray whatever of the ]

second part, i

the correspondmg points of the segment AB determine two parts of the i

segments such that \

ii) every point of the segment AB belongs to one of the two parts, )

2^ the extremity A belongs to the first part and B to the second,
3) any point whatever of the first part precedes any point whatever of ';
the second. I

But in that case there exists a point C of AB (which may belong to one
or the other of the two parts) such that every point of AB that precedes C
belongs to the first part and every point of AB that follows C belongs to the
second part

I

s

I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 237

Thus exactly the same thing holds of Cy the ray corresponding to C, with
reference to the division of the angle (ab) into two parts.

It is not difficult to extend this to an angle (ab) which is either flat or
greater than two right angles ; this is done (Bonola, op, cit. pp. 87 — 88) by
supposing the angle to be divided into two, (ad)y {db)^ each less than two
ri^t angles, and considering the three cases in which

(i) Sie ray ^ is such that all the rays that precede it belong to the first
part and those which follow it to the second part,
the ray ^ is followed by some rays of the first part,
the ray d is preceded by some rays of the second part

Application to circular arcs.

If we consider an arc AB of a circle with centre Oy the points of the arc
correspond univocally, and in the same order, to the rays from the point O
passing through those points respectively, and the same argument by which
we pa^ed from a segment of a straight line to an angle can be used to make
the transition from an angle to an arc.

Intersections of a straight line with a circle.
It is possible to use the Postulate of Dedekind to prove that

If a straight line has one point inside and one point outside a circle^ it has
two points common with the circle.

For this purpose it is necessary to assume (i) the proposition with reference
to the perpendicular and obliques drawn from a given point to a given straight
line, namely that of all straight lines drawn from a given point to a given
straight line the perpendicular is the shortest, and of the rest (the obliques)
that is the longer which has the longer projection upon the straight line, while
those are eqiud the projections of which are equal, so that for any given
length of projection there are two equal oblic^ues and two only, one on each
side of the perpendicular, and (2) the proposition that any side of a triangle
is less than the sum of the other two.

Consider the circle (C) with centre Oy and a straight line (r) with one
point A inside and one point B outside the
circle.
) By the definition of the circle, if J? is /^ p AJJK B

the radius,

OA<Ry OB>R.

Draw OP perpendicular to the straight
line r.

Then 0P< OAy so that OP is always
less than J?, and P is therefore within the
circle C

Now let us fix our attention on the finite segment AB of the straight
line r. It can be divided into two parts, (i) that containing all the points H
for which OH<R (i.e. points inside C), and (2) that containing all the
points K for which OK ^ R (points outside C or on the circumference of C).
m Thus, remembering that, of two obliques from a given point to a given
straight line, that is greater the projection of which is greater, we can assert
that all the points of the segment PB which precede a point inside C are
inside C, and those ^iYa^ follow a point on the circumference of C or outside
t C are outside C.

Hence, by the Postulate of Dedekind, there exists on the segment PB a

f

238 ROOK I [1. Axx.

point Jf such that all the points which precede it belong to the fint part and
those which follow it to the second part

I say that Afis common to the straight line rand the circle C, or

For suppose, e.g., that OJIf< R.

There will then exist a segment (or length) <r less than the difference
between R and OM.

Consider the point M*^ one of those which foUcw M^ such that MMT is
equal to <r.

Then, because any side of a triangle is less than the sum of the other two,
OM' < OM^ MMT.

But OM^MM'^OM-¥ir<X,

whence OM'<X^

which is absurd.

A similar absurdity would follow if we suppose that OM > X.

Therefore OM must be equal to R.

It is immediately obvious that, corresponding to the point if on the segment
F£ which is common to r and C, there is another pomt on r which. has the
same property, namely that which is symoaetrical to AT with respect to P.

And Uie proposition is proved

Intersections of two circles.

We can likewise use the Postulate of Dedekind to prove that

If in a givtn plane a drde C has onepoint X inside and^nepaini Y auisUe
another drde C, the two cirdes intersect in two points.

We must first prove the following

Lemma,

If Oj O' are the centres of two circles (7, C\ and J?, R' their radii
respectively, the straight line 00' meets the circle C in two points A^ B^ on6
of which is inside C and the other outside it ^

Now one of these points must fall (i) on the prolongation of ffO
beyond O or {2) on Off itself or (3) on the (

prolongation of Off beyond ff.

(i) First, suppose A to lie on ffO pro-
duced.

Then Aff^AO-^OO'-^R^ Off (a).

But, in the triangle Off V,

ffY<OY^Off,
and, since ffY>R, OY^R,
R<R^Off.

It follows from M that Aff>R\ and A
therefore lies outside C,

(2) Secondly, suppose A to lie on
Off.

Then Off ^OA^Aff^R^Aff ,,.{fi).
From the triangle OffX we have b

Off^OX^ffX,
and, since OX^R^ ffX<R^ it follows
that

Off<R^R,
whence, by {fi\ Aff <R,so that A lies inside C.

I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 239

(3) Thirdly, suppose A to He on OO produced.

Then R^OA^OO^OA (y).

And, in the triangle OOX^

OX < 00^ OX, Y/

thatb R<.0O^^(yX, ^\

whence, by (y),

oo^o'A<oa^ox, c\

or O'A < ax,

and A lies inside C

It is to be observed that one of the two points A, B is in the position of
case (i) and the other in the position of either case (2) or case (3) : whence
we must conclude that one of the two points ^, ^ is inside and the other
outside the circle C\

Proof of theorem.

The circle C is divided by the points A, B into two semicircles. Consider
one of them, and suppose it to be
described by a point moving from A
to B.^

Take two separate points F, Q
on it and, to fix our ideas, suppose
that P precedes Q.

Comparing the triangles OOP,
OOQymt observe that one side OO
is common, OP is equal to OQ, and
the angle POO is less than the angle
QOO.

Therefore OP<OQ.

Now, considering the semicircle APQB as divided into two parts, so that
the points of the first part are inside the circle C\ and those of the second
part on the circumference of C or outside it, we have the conditions necessary
for the applicability of the Postulate of Dedekind (which is true for arcs of
circles as for straight lines) ; whence there exists a point M separating ttu two
parts,
' I say that OM^ R.

\ For, if not, suppose OM < R!,

ilf then o- signifies the difference between R and OM, suppose a point M\
which follows M, taken on the semicircle such that the chord MM' is not
greater than o- (for a way of doing this see below).
^ Then, in the triangle OMM*,

OM* < OM^ MM* < OM-¥ <r,
and therefore OM' < R.

It follows that AT, a point on the arc MB, is inside the circle C :
which is absurd.

Similarly it may be proved that OMis not greater than R.
Hence OM=R.

[To find a point M such that the chord MM* is not greater than <r, we
may proceed thus.

Draw from Mb, straight line MP distinct from OM, and cut off MP on it
equal to v/i.

240

BOOK I

[i. Axx.

Join OP^ and draw another radius OQ such that the ang^e POQ is equal
to the angle MOP. q

The intersection, M\ of OQ with the
circle satisfies the required condition.

For MM' meets OP at right angles
in 5.

Therefore, in the right-ai^led triangle
MSP^ MS is not greater than MP (it is
less, unless MP coincides with if5, when
it is equal).

Therefore MS is not greater than <r/2, so that MMT is not greater than v.]

J

i i

I

I

!(

ft ■

B(X)K I. PROPOSITIONS.

Proposition i.

I On a given finite straight line to construct an equilateral

triangle.

I' Let AB be the given finite straight line.

* Thus it is required to con-

fS struct an equilateral triangle on
' the straight line A£.

With centre A and distance
AB let the circle BCD be
described ; [Post 3]

'^o again, with centre B and dis-
1 tance BA let the circle ACE
I be described ; [Post 3]

:. and from the point C, in which the circles cut one another, to
\ the points -/4, B let the straight lines CA^ CB be joined.
\ [Post. I]

15 Now, since the point A is the centre of the circle CDB^
j AC is equal to AB. [Def. 15]

Again, since the point B is the centre of the circle CA£y
J i BC is equal to BA. [Def. 15]

But CA was also proved equal to AB ;

iao therefore each of the straight lines CA, CB is equal to AB.

And things which are equal to the same thing are also

equal to one another ; [C. N. i]

therefore CA is also equal to CB.

Therefore the three straight lines CA^ AB^ BC are
as equal to one another.

H. E. 16

BOOK I [1.1

Therefore the triangle ABC is equilateral ; and it has
been constructed on the given finite straight line AB.

(Being) what it was required to do.

I. On a given finite straight line. The Greek usage diffen finom oars in that the
definite article is employed in sudi a phrase as this where we have the indefinite. M rft
do$€lfffit c^ffittf w€W€pafffUpiit, **on iJkg given finite straight line," i.e. the finite straight line
which we choose to take.

3. Let AB be the given finite straight line. To be strictly literal we should have to
translate in the reverse order "let the given finite straight line be the (straight line) AB^i
bat this order is inconvenient in other cases where there is more than one datam, e^ in the
sttting-aut of I. 3, *'let the given point be A^ and the given straight line BC^** tihe awkward-
ness arising from the omission of the verb in the second daose. Hence I have, for deamess'
sake, adopted the other order throagfaoat the book.

8. let the circle BCD be described. Two things are here to be noted, (1) the elegant
and practically universal use of the perfect passive imperative in constructions, yrfp £ f $t »
meaning of course "let it Aatfe heen described " or "suppose it described,** (3) the impossi-
bility o7 expressing shortly in a translation the force of the words in their original order.
jci^irXot ytypi^ta 6 BFA means literally "let a circle have been described, the (circle, namehr,
which I denote bv) BCD" Similarly we have lower down *' let straight lines, (namely) the
(straight lines) CA, CB, be joined,'* iwttfB^Stfaap MtUu td FA, r£ There aecms to be
no practicable alternative, in En^i^ bat to translate as I have done in the teat.

13. from the point C... Eodid is carefid to adhere to the phraseoloMpr of Pottalate i
except that he speaks of "joining" (^rc^r^dMvar) instead of " drawing '^(W^^tiv). He
does not allow himself to use the shortenei expression " let the straight line r^ be joined "
(without mention of the points /*, C) until i. 5.

10. each of the straight lines CA, CB, inripa rflp TA, FB and 94. the three
straight lines CA, AB, BC, ol rpeit al FA, AB, BF. I have, here and in all similar
expressions, inserted the words "straight lines" which are not in the Greek. The possettion
of the inflected definite article enables the Greek to omit the words, but tlds is not possible
in English, and it would scarcely be Ei^lish to write "each of CA, CB" or "the three CA,

AB, Bc:'

It is a commonplace that Euclid has no right to assume, without pre-
mising some postulate, that the two circles wtJl meet in a point C To
supply what is wanted we must invoke the Principle of Continuity (see note
thereon above, p. 235). It is sufficient for the purpose of this proposition and
of I. 22, where there is a similar tacit assumption, to use the form of postulate
suggested by Killing, ''/if ^ ^ne [in this case e.g^ the circumference AC£]
belongs entirely to a figure [in this case a plane] which is divided into two parts
[namely the part enclosed within the circumference of the circle BCD and
the part outside that circle], and if the line has at least one point common with
each part, it must also meet the boundary between the parts [Le. the circum-
ference ACE must meet the circumference BCLf\,^^

Zeno's remark that the problem is not solved unless it is taken for granted
that two straight lines cannot have a common s^ment has already been
mentioned (note on Post 2, p. 196). Thus, if AC, BC meet at F before
reaching C, and have the part FC common, the triangle obtained, namely
FAB, will not be equilateral, but FA, FB will each be less than AB, But
Post 2 has already laid it down that two straight lines cannot have a common
segment

Proclus devotes considerable space to this part of Zeno's criticism, but
satisfies himself with the bare mention of the other part, to the effect that it
is also necessary to assume that two circumferences (with different centres)
cannot have a common part That is, for anything we know, there may be
any number of points C common to the two circumferences ACE, BCD. It
is not until in. 10 that it is proved that two circles cannot intersect in more

<

il

1. 1]

PROPOSITION I

243

points than two, so that we are not entitled to assume it here. The most we
can say is that it is enough for the purpose of this proposition if one equilateral
triangle can be found with the given base ; that the construction only gives
two such triangles has to be left over to be proved subsequently. And indeed
we have not long to wait ; for i. 7 clearly shows that on either side of the
base AB only one equilateral triangle can be described. Thus i. 7 gives us
the numlfer of solutions of which t^e present problem is susceptible, and it
supplies the same want in i. 22 where a triangle has to be described with
three sides of given length ; that is, i. 7 furnishes us, in both cases, with one
of the essential parts of a complete Siapur|A<^ which includes not only the
determination of the conditions of possibility but also the number of solutions
(voo-axolf ^x^P<^ Proclus, p. 202, 5). This view of i. 7 as supplying an
equivalent for iii. 10 absolutely needed in i. i and i. 22 should serve to correct
the idea so common among writers of text-books that i. 7 is merely of use as a
lemma to Euclid's proof of i. 8, and therefore may be left out if an alternative
proof of that proposition is adopted.

Agreeably to his notion that it is from i. i that we must satisfy ourselves
that isosceles and scalene triangles actually exist, as well as equilateral triangles,
Proclus shows how to draw, first a particular isosceles triangle, and then a
scalene triangle, by means of the figure of the proposition. To make an
isosceles triangle he produces AB in both directions to meet the respective
circles in Z?, E^ and then describes
circles with A^ B ais centres and AE^
BD as radii respectively. The result is
an isosceles triangle with each of two
sides double of the third side. To make
an isosceles triangle in which the equal
sides are not so related to the third side
but have any given length would require
. the use of i. 3 ; and there is no object in
treating the question at all in advance of
I. 22. An easier way of satisfying our-
selves of the existence of some isosceles
triangles would surely be to conceive any
two radii of a circle drawn and their
extremities joined.

There is more point in Proclus* construction of a scalene triangle. Suppose
^C to be a radius of one of the two
circles, and D a point on AC lying in
that portion of the circle with centre A
which is outside the circle with centre B,
Then, joining BD^ as in the figure, we
have a triangle which obviously has all its
sides unequal, that is, a scalene triangle.

The above two constructions appear in
an-Nairlzfs commentary under the name
of Heron; Proclus does not mention his
source.

In addition to the above construction
for a scalene triangle (producing a triangle in which the ''given" side is
greater than one and less than the other of the two remaining sides), Heron
has two others showing the other two possible cases, in whic^ the '' given "
side is (i ) less than, (2) greater than, either of the other two sides.

16 — 2

UA

BOOR I
Proposition 2.

[Postal

To place at a given paint (as an extremity) a straight Une
equal to a given straight line.

Let A be the given point, and BC the given straight line.
Thus it is required to place at the point A (as an extremity)
5 a straight line equal to the given
straight line BC.

From the point A to the point B
let the straight line AB be joined ;

[Post i]

and on it let the equilateral triangle

JO DAB be constructed. [i. i]

Let the straight lines AE, BF be

produced in a straight line with DA^

DB\ [Post a]

with centre B and distance BC let the

15 circle CGH be described ;

and again, with centre D and distance DG let the circle GKL

be described. [Post 3]

Then, since the point B is the centre of the circle CGH^

BC is equal to BG.

so Again, since the point D is the centre of the circle GKL,

DL is equal to DG.
And in these DA is equal to DB ;

therefore the remainder AL\s equal to the remainder
BG. \C.N. 3]

But BC was also proved equal to BG ;

therefore each of the straight lines AL, BC is equal
\oBG.
And things which are equal to the same thing are also
equal to one another ; [C.AI i]

J therefore AL is also equal to BC.

Therefore at the given point A the straight line AL is
placed equal to the given straight line BC.

(Being) what it was required to do.

I. (as an extremity). I have inserted these words because **to place a stiaisht line
at a given point '* (vp^t r^ Ba$4vri ^Mc(^)is not quite clear enough, at least in English.

la Let the straight lines AE, BP be produced.... It will be observed ^t in this
first application of Postulate 3, and a^ ain in i. 5, Euclid speaks of the amHnuaiiom ci the
straight Une as that which is produced in such cases, iKfi€p\^waaif and wpoatxPtfiK^^Sttntw
meaninglittle more than drawing straight lines " in a straight line with " the given straight
lines. The first place in which Euclid uses phraseology exactly corresponding to onn when

as

I

r

1.2] PROPOSITION 2 24s

speakii^ of a straight line being produced is in i. 16 :" let one tide of it, BC^ be produced

to Z> " {wpofftKPtpX^Bw a^oO fda, wXevpii iiBT iwlrb A).

22. the remainder AL...the remainder BO. The Greek expressions are Xocr^ ^
j AA and Xocrj rv BH, and the literal translation would be '*AL (or BG) remaining,**
! but the shade of meaning conveyed by the position of the definite article can hardly be

expressed in English.

This proposition gives Proclus an opportunity, such as the Greek
\ commentators revelled in, of distinguishing a multitude of ^ases. After
I explaining that those theorems and problems are said to have cases which

• have the same force, though admitting of a number of different figures, and
t preserve the same method of demonstration while admitting variations of
^ position, and that cases reveal themselves in the construction^ he proceeds to

distinguish the cases in this problem arising from the different positions

which the given point may occupy relatively to the given straight line. It may

be (he says) either (i) outside the line or (2) on the line, and, if {i\ it may be

\ either {a) on the line produced or {b) situated obliquely with regard to it ; if

. (2), it may be either {a) one of the extremities of the line or {b) an intermediate

> point on it It will be seen that Proclus* anxiety to subdivide leads him to

I give a "case," (2) (a), which is useless, since in that "case" we are given

what we are re(}uired to find, and there is really no problem to solve. As

I Savile says, " qui quaerit ad p punctum ponere rectam aequalem r^ fiy rectae,

j quaerit quod datum est, quod nemo faceret nisi forte insaniat"

! Proclus gives the construction for (2) (b) following Euclid's way of taking

t (? as the point in which the circle with centre B intersects DB produced^ and

then proceeds to " cases," of which there are still more, which result from the

different ways of drawing the equilateral triangle and of producing its sides.

• This last class of " cases " he subdivides into three according as AB is
(i) equal to, (2) greater than or (3) less than BC, Here again " case " (i) serves
no purpose, since, if AB is equal to BC^ the problem is already solved. But
Proclus' figures for the other two cases are worth giving, because in one of
them the point G is on BD produced beyond Z>, and in the other it lies on
BD itself and there is no need to produce any side of the equilateral triangle.

A glance at these figures will show that, if they were used in the proposition,
each of t}iem would require a slight modification in the wording (i) of the
construction, since BD is in one case produced beyond D instead of B and
in the other case not produced at all, (2) of the proo^ since BG, instead of
being the difference between DG and DBy is in one case the sum of DG and
DB and in the other the difference between DB and DG.

346 BOOK I

Modem editors generaUy seem to dassify the cues aooofding to Ae
possible variations in the construction rather than according to differences in
the data. Thus Lardner, Potts, and Todhunter distinguiui eight cases due
to the three possible alternatives, (i) that the given point may be joined to
either end of the given straight line, (2) that the equilateral triangle may then
be described on either side of the joining line, and (3) that the side of the
equilateral triangle which is produced may be produced in either direction.
(But it should Imve been observed that, where AB is greater than BC^ tfie
third alternative is between producing DB and not producing it at all) Potts
adds that, when the given pomt lies either on the line or on the line pioduoed,
the distinction which arises from joining the two ends of the line with the
given point no longer exists, and there are only four cases of the problem
(I think he should rather have said sohiHons).

To distinguish a number of cases in this way was foreign to the really
classical manner. Thus, as we shall see, Euclid's method is to give one case
only, for choice the most difficult, leaving the reader to suf^ly the rest for
himself. Where there was a real distinction between cases, sufficient to
necessitate a substantial difference in the proof, the practice was to gjve
separate enunciations and -proofs altogether, as we may see, e.g., from the
Conies and the De sectione ratioms of Apollonius.

Proclus alludes, in conclusion, to the error of those who proposed to solve
I. 2 by describing a circle with the given point as centre and with a distance
equcU to J9C, which, as he says, is a ptHtio prindpH. De Morgan puts the
matter very clearly {Supplementary Remarks on the first six Books of EudS^s
Elements in the Companion to the Almanac^ 1349, P- 6). We should ^'insist,"
he says, ''here upon the restrictions imposed by the first three postulat^
which do not allow a circle to be drawn with a compass-carried distance;
suppose the compasses to close of themselves the moment they cease to toudi
the paper. These two propositions [i. 2, 3] extend the power of construction
to what it would have been if all the usual power of the compasses had been
assumed ; they are mysterious to all who do not see that postulate iii does
not ask for every use of the compasses P

Proposition 3.

Given two unequal straight lines^ to cut off front the
greater a straight line equal to the
less. 5.

Let ABy C be the two given un-
equal straight lines, and let AB be
the greater of them.

Thus it is required to cut off from
AB the greater a straight line equal
to C the less.

At the point A let AD be placed
equal to the straight line C ; [i. 2]
and with centre A and distance AD let the circle DEF be
described. [Post 3]

*\

!

1.3,4] PROPOSITIONS 2—4 247

Now, since the point A is the centre of the circle DEF,

AE is equal to AD. [Def. 15]

But C is also equal to AD.

Therefore each of the straight lines AE, C is equal to
AD ; so that AE\s also equal to C. \C.N, i]

Therefore, given the two straight lines AB, C, from AB
the greater A[E has been cut off equal to C the less.

(Being) what it was required to do.

Proclus contrives to make a number of ''cases" out of this proposition
also, and gives as many as eight figures. But he only produces this variety by
practically incorporating the construction of the preceding proposition, instead
of assuming it as we are entitled to do. If Prop. 2 is assumed, there is really
only one " case " of the present proposition, for Potts' distinction between two
cases according to the particular extremity of the straight line from which the
given length has to be cut off scarcely seems to be worth making.

Proposition 4.

If two triangles have the two sides equal to two sides
respectively, and have the angles contained by the equal straight
lines equal, they will also have the base equal to the base, the
triangle will be equal to the triangle, and the remaining angles
5 will be equal to the remaining angles respectively, namely those
which the equal sides subtend

Let ABC, DEE he two triangles having the two sides
AB, AC equal to the two sides DE, Z?-F respectively, namely
AB to DE and AC to DE, and the angle BAC equal to the
ID angle EDE.

I say that the base BC is also equal to the base EE, the
triangle ABC will be equal to the triangle DEE, and the
remaining angles will be equal to the remaining angles
respectively, namely those which the equal sides subtend, that
15 is, the angle ABC to the angle DEE, and the angle ACB
to the angle DEE.

For, if the triangle ABC be
applied to the triangle DEE,
and if the point A be placed
20 on the point D

and the straight line AB
on DE,

then the point B will also coincide with E, because AB is
equal to DE.

348 BOOK I [1.4

5 Again, AB coinciding with DE^

the straight line AC will also coincide with DF, because the

angle BAC is equal to the angle EDF\

hence the point C will also coincide with the point F^

because AC\s again equal to DF.
o But B also coincided with E ;

hence the base BC will coincide with the base EF.

[For if, when B coincides with E and C with F, the base

BC does not coincide with the base EF, two straight lines

will enclose a space : which is impossible.

5 Therefore the base BC will coincide with

EF^ and will be equal to it \C.N. 4]

Thus the whole triangle ABC will coincide with the
whole triangle DEF,

and will be equal to it.

p And the remaining angles will also coincide with the
remaining angles and will be equal to them,

the angle ABC to the angle DEF,

and the angle ACB to the angle DFE.

Therefore etc.
^5 (Being) what it was required to prove

I — 3. It is a fiict that Eadid*s enunciations not infreauently leave something to be
desued in point of clearness and precision. Here he speaks of the triangles having "the
angle equal to the angle, namely the angle contained by the equal straight lines ** (rV Ttfrfor
ri yttwlg, Uniff fxtl '''¥ ^ f^ ^^^ t^tiQ^ vcptcxo/i^nyr), only one of the two angles being
described in the latter expression ^n the accusative), and a similar expression in the dative
being left to be understood of the other angle. It is curious too that, after mentioning two
"sides" he speaks of the angles contained by the equal ** straight lines," not ^'sidesy It
majr be that ne wished to i^ere scrupulously, at the outset, to the phraseolof^ of the
dennitions, where the angle is the inclination to one another of two iines or strait lines.
Similarljr in the enunciation of i. 5 he speaks of producing the equal '* straight lines " as if to
keep stnctly to the wording of Postulate 3.

3. respectively. I agree with Mr H. M. Taylor {Euclid, p. ix) that it is best to
abandon the traditional translation of "each to each,' which would naturally seem to imply
that all the four magnitudes are equal rather than (as the Greek iKaHpa ixaiip^ does) tnat
one is eoual to one and the other to the other.

3. tne base. Here we have the word ^au used for the first time in the Elements,
Proclus explains it (p. 336, 13 — 15) as meaning (i), when no side of a triangle has been
mentioned oefore, the side ** which is on a level with the sight *' (rV rp^ rg 6^ K»fi4nfw),
and (3), when two sides have already been mentioned, the third side. Ftodus thus avoids
the mistake made by some modem editors who explain the term exclusively with reference to
the case where two sides have been mentioned before. That this is an error is proved (i) by
the occurrence of the term in the enunciations of i. 37 etc. about triangles on tne same base
and equal bases, (3) by the application of the same term to the bases of parallelograms in
I. 35 etc. The truth is that the use of the term must have been suggested oy the practice of
drawing the particular side horizontally, as it were, and the rest of the figure above it. The
dose of a figure was therefore spoken of, primarily, in the same sense as ue base of anything

■(

J

!

1.4] PROPOSITION 4 249

else, e.g. of a pedestal or column; but when, as in i. 5, two triangles were compared
occupying other than the normal positions which gave rise to the name, and when two sides
had been previously mentioned, the base was, as Proclus says, necessarily the third side.

6. subtend, itwvrtlwtup ifw6, **to stretch under," with accusative.

9. the angle B AC. The fiill Greek expression would be ^ ^6 rwv BA, AF v€pux90uhii
YMrto, "the angle contained by the (straight lines) BjI^ AC." But it was a common practice
of Greek geometers, e.g. of Archimedes and Apollonius (though not apparently of Euclid), to
use the abbreviation «l BAP for «l BA, AF, **the (straight lines) BA, AC.** Thus, on
«-cptcxoA</n| being dropped, the expression would become first ^ M rtDv BAT Tw^fo, then
4 ^6 BAT ywla, and finally ^ inr6 BAP, without ytatda, as we rqrolarly find it in Euclid.

17. if the triangle be applied to..., 33. coincide. The difference between the
technical use of the passive i^apfU^taBcu "to be a^piitd {to)" and of the active i^apfUftuf
"to eeindde (with) '^ has been noticed above (note on Common Notion 4, pp. 334 — 5).

J 3. [For \i, when B coincides... ^6. coincide with EF]. Heiberg {P^troHpcminmau
fid in Hermis, xxxviii., 1903, p. 50) has pointed out, as a conclusive reason for regarding
these words as an early interpolation, that the text of an-Nairld {Codex Leidensis ^99, 1, ed.
Besthom-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.B.D.,
which shows that they constitute a scholium only. They were doubtless added bjjr some
commentator who thought it necessary to explain the immediate inference that, smce B
coincides with E and C with F, the straight line BC coincides with the straight line RF,
an inference which really follows from the definition of a straight line and Post. 1 ; and no
doubt the Postulate that " Two straight lines cannot enclose a space " (afterwards placed
among the Common Notions) was interpolated at the same time.

44. Therefore etc. Where (as here) Euclid's conclusion merely repeats the enunciation
( word for word, I shall avoid the repetition and write " Therefore etc.*' simply.

I In the note on Common Notion 4 I have already mentioned that Eudid

^ obviously used the method of superposition with reluctance, and I have given,

after Veronese for the most part, the reason for holding that that method is

not admissible as a theoretical means of proving equality, although it may be

of use as 9^ practical test, and may thus furnish an empirical basis on which to

found a postulate. Mr Bertrand Russell observes {Principles of Maihemaiics

I. p. 405) that Euclid would have done better to assume i. 4 as an axiom, as

. is practically done by Hilbert {Grundlagen der Geometric^ p. 9). It may be

( / that Euclid himself was as well aware of the objections to the method as are

i ' his modem critics ; but at all events those objections were stated, with almost

\^ equal clearness, as early as the middle of the i6th century. Peletarius

(Jacques Peletier) has a long note on this proposition (In EucHdis EUmemta

j geometrica demonstrationum libri sex^ i557)> in which he observes that, if

I' superposition of lines and figures could be assumed as a method of proof^ the

\ whole of geometry would be full of such proofs, that it could equally well have

4 been used in i. 2, 3 (thus in i. 2 we could simply have supposed the line taken

s up zxA placed at the point), and that in short it is obvious how far removed the

' method is from the dignity of geometry. The. theorem, he adds, is obvious in

itself and does not require proof ; although it is introduced as a theorem, it

would seem that Euclid intended it rather as a definition than a theorem, '^for

I cannot think that two angles are egual unless I have a conception of what

equality of angles is.'' Why then did Euclid include the proposition among

theorems, instead of placing it among the axioms ? Peletanus makes the best

excuse he can, but concludes thus: '' Huius itaque propositionis veritatem non

aliunde quam a communi iudido petemus; cogitabimusque figuras figuris

superponere, Mechanicum quippiam esse: intelUgere verb, id demum esse

Mathemadcum.''

Expressed in terms of the modem systems of Congruence-Axioms referred
to in the note on Common Notion 4, what Euclid really assumes amounts to
»' the following :

(i) On the line DE^ there is a point E^ on either side of 27, such that AB
is equal to DE.

i

250 BOOK I [1.4

(2) On either side of the ray DE there is a ray DF such that the angle
EDF'v^ equal to the angle BAC.

It now follows that on DF there is a pcnnt F such that DF is equal
to^C.

And lastly (3), we require an axiom from which to infer that the two
remaining angles of the triangles are respectively equal and that the bases aie
equal.

I have shown above (pp. 229—330) that Hilbert has an axiom stating the
equality of the remaining angles simply, but proves the equality of the b^^ses.

Another alternative is that of Pasdi ( Variesungm uber neuert Geomdrk^
p. 109) who has the following ''Grundsatz":

If two figures AB and FGH are given (FGH not being contained in a
straight length), and AB^ FG are congruent and if a plane surfoce be laid
through A and B^ we can specify in this plane surface, produced if necessary,
two points C, Z>, neither more nor less, such that the figures ABC and ABD
are congruent with the figure FGH^ and the straight Tine CD has with the
straight line AB or with AB produced one point common.

I pass to two points of detail in Euclid's proof:

(i) The inference that, since B coincides with E^ and C with F^ the
bases of the triangles are wholly coincident rests, as expressly stated, on the
impossibility of two straight lines enclosing a space, and therefore presents no
difficulty.

But (2) most editors seem to have fiuled to observe that at the very
beginning of the proof a much more serious assumption is made without any
explanation whatever, namely that, if ^^ be placed on D^ and AB on DE^ the
pomt B will coincide with ^, because AB is equal to DE. That is, the
converse of Common Notion 4 is assumed for straight lines. Produs merely
observes, with regard to the converse of this Common Notion, that it is only
true in the case of things "of the same form " (6/ioc(S^), which he explains as
meaning stnught lines, arcs of one and the same circle, and angles '' contained
by lines simiku: and similarly situated" (p. 241, 3 — 8^.

Savile however saw the difficulty and grappled with it in his note on the
Common Notion. After stating that all straight lines with two points common
are congruent between them (for otherwise two straight lines would enclose a
space), he argues thus. Let there be two straight lines AB^ DE^ and let A be
placed on 2?, and AB on DE. Then B will coincide with E. For, if not,
let B fall somewhere short of E or beyond E ; and in either case it will follow
that the less is equal to the greater, which is impossible.

Savile seems to assume (and so apparently does Lardner who gives the
same proof) that, if the straight lines be "applied," B will fall somewhere on
DE or DE produced. But the ground for this assumption should surely be
stated ; and it seems to me that it is necessary to use, not Postulate i alone,
nor Postulate 2 alone, but both, for this purpose (in other words to assume,
not only that two straight lines cannot enclose a space^ but also that two straight
lines cannot have a common segment). For the only safe course is to place A
upon D and then turn AB about D until some point on AB intermeduite
between A and B coincides with some point on DE. In this position AB an^
DE have two points common. Then Postulate i enables us to infer that the
straight lines coincide between the two common points, and Postulate 2 that
they coincide beyond the second common point towards B and E. Thus the
straight lines coincide throughout so fieur as both extend; and Savile's argument
then proves that B coincides with E.

J

i.,5] PROPOSITIONS 4, 5 2^1

Proposition 5.

In isosceles triangles the angles at the base are equal to one
another^ and, if the equal straight lines be produced further,
the angles under the base will be equal to one another.

Let ABC be an isosceles triangle having the side AB
5 equal to the side AC\

and let the straight lines BD, CE be produced further in a
straight line with AB, AC. [Post 2]

1 say that the angle ABC is equal to the angle ACB^ and
the angle CBD to the angle BCE.
10 Let a point F be taken at random
on BD]

from AE the greater let AG he cut off
equal to AF the less ; [i. 3]

and let the straight lines FC, GB be joined.

[Post i]

15 Then, since AF is equal to -^G^ and
^^to^C,

the two sides FA^ AC are equal to the
two sides GA, A By respectively ;
and they contain a common angle, the angle FAG.
20 Therefore the base FC is equal to the base GB,

and the triangle AFC is equal to the triangle AGB,
and the remaining angles will be equal to the remaining angles
respectively, namely those which the equal sides subtend,

that is, the angle ACF to the angle ABG,
25 and the angle AFC to the angle AGB. [i. 4]

And, since the whole AFis equal to the whole AG,
and in these AB is equal to AC,
the remainder BF is equal to the remainder CG.
But FC was also proved equal to GB ;
30 therefore the two sides BF, FC are equal to the two sides
CG, GB respectively ;
and the angle BFC is equal to the angle CGB,

while the base BC is common to them ;
therefore the triangle BFC is also equal to the triangle CGB,
35 and the remaining angles will be equal to the remaining

[

353 BOOK I [l 5

angles respectively, namely those which the equal sides
subtend ;

therefore the angle FBC is equal to the angle GCB^
and the angle BCF to the angle CBG.
¥> Accordingly, since the whole angle ABG was proved
equal to the angle ACF,

and in these the angle CBG is equal to the angle BCF,

the remaining angle ABC is equal to the remaining angle
ACB;

15 and they are at the base of the triangle ABC.

But the angle FBC was also proved equal to the angle GCB ;

and they are under the base.

Therefore etc. Q. E. d.

1. the equal straight lines (meaniiig the equal sides). Cf. note on the amilar
expression in Frop. 4, lines 1, $.

la Let a point P be taken at random on BD, c2X4^^ M rft BA rvx^ tfit^wSbr H E,
where rvx^v aii/uw means "a chance point.**

17. the two sides FA, AC are equal to the two sides QA, AB respccthreljr, Mo
ol Zl, AT ival nut HA, AB Krai tlaiw ixmripa inrip^. Here, uid in nnmberlest later
passages, I have inserted the word "ades** for the reason given in the note on I. i, fine so.
It would have been permissible to sopply either '* straight lines" or "ades**; but on the
whole "sides '* seems to be more in accordance with the phrasedogjr of I. 4.

33. the base BC is common to them, i.e., apparently, common to the atngUSf as
the odrc^v in fidatt ai^rwr icoirii can only refer to ytpUt and yvAa preceding. Simson wrote
**and the base ^C is common to the two triangles BFC^ CGB*'\ Todhunter left out these
words as being of no use and tending to perplex a beginner. But Euclid evidently chose
to quote the conclusion of I. 4 exactly ; the first phrase of that conclusion is that the bases
(of the two triangles) are equal, and, as the equal bases are here the sams base, Euclid
naturally substitutes the word "common** for "equal."

48. As ** (deing) what it was required to prove '* (or " do '*) is somewhat long, I shall
henceforth write the time-honoured "Q. E. D.' and "Q. E. F." for Swtp idu 9€t^ and 9rtp

According to Proclus (p. 250, 20) the discoverer of the fact that in any
isosceles triangle the angles at the base are equal was Thales, who however
is said to have spoken of the angles as being similar^ and not as being equal,
(Cf. Arist De caelo iv. 4, 31 1 b 34 irpos hyjom -^^wm ^v€rai ^cpoficyoy where
egua/ angles are meant.)

A pre-Euclidean proof of I. 5.

One of the most interesting of the passages in Aristotle indicating differences
between Euclid's proofs and those with which Aristotle was fieunilmr, in other
words, those of the text-books immediately preceding Euclid's, has reference to
the theorem of i. 5. The pissage (Anal. Prior, i. 24, 41 b 13—22) is so
important that I must quote it in full. Aristotle is illustrating the fact that in
any syllogism one of the propositions must be affirmative and universal
(ffoMXov). ''This," be says, ''is better shown in the case of geometrical
propositions ^ i^ roTs &aypafi/Aao-tv), e.g. the proposition that the angles at the
base of an isosceles triangle are equal,

"For let ^, ^ be drawn [i.e. joined] to the centre.

I. S] PROPOSITION 5 «53

''If, then, we assumed (i) that the angle AC [Le. ^ + C] is equal to die
angle BD [Le. B-^D^ without asserting generally
that the angles of semidrcUs are equals and again
(2) that the angle C is equal to the angle D without
making the further assumption that the two angles of
ali segments are equals and if we then inferred, lastly,
that, since the whole angles are equal, and equal
angles are subtracted from them, the angles which
remain, namely ^, ^ are equal, we should commit
a petitio prinapii'^

Some peculiarities of phraseology will be observed
in this passage.

(i) A^ B are said to be drawn (4yfiiyai) to the centre (of the cirde of
which the two equal sides are radii) asif A^B were not the angular points but
the sides or the radii themselves. There is at least one pandlel for this in
Euclid (cf. IV. 4).

(2) "The angle AC*' is the angle which is the sum of A and C, and A
means here the angle at A of the isosceles triangle shown in the figure, and
afterwards spoken of by Aristotle as £, while C is the " mixed " angle between
AB and the circumference of the smaller segment cut off by it

(3) The ''angle of a semicircle" (i.e. the "angle" between the diameter
a^d the circumference, at the extremity of the diameter) and the "angle ^ a
segment" appear in Euclid in. 16 and in. Def. 7 respectively, obviously as
survivals from earlier text-books. ■ .^.-^

But the most significant facts to be gathered from the extract are that in
the text-books whidb preceded Euclid's " mixed " angles played a much more

important part than they do with Euclid, and, in particular, that at least two "^

propositions concerning such angles appeared quite at the beginning, namdy "x

the propositions that the (mixed) angles of semicircles are equal and that the two .%

(fkixed) angles of any segment of a circle are equal. The wording of the first . ;: ;

of the two propositions is vague, but it does not necessarily mean more dian ' V-^

that the two (mixed) angles in one semicircle are equal, and I know of no -J;

evidence going to show that it asserts that the angle of any one semicircle is ^^
equal to the angle of any other semicircle (of different size). It is quoted in _ "^

the same form, " because the angles of semicircles are equal," in die Latin n^

translation from the Arabic of Heron's Catoptrica^ Prop. 9 (Heron, VoL VL^ '^

Teubner, p. 334), but it is only inferred that the different radii of one drde *^ V?

make equal "angles" with the circumference ; and in the similar proposition --^)

of the Pseudo-Euclidean Catcptrica (Euclid, Vol. vii., p. 294) ai4;les of the >

same sort in one circle are said to be equal "because they are (angles) of "'^

a semicircle." Therefore the first of the two propositions may be cmly a ' \

particular case of the second. . t>

But it is remarkable enough that the second proposition (that the two "^

" CMgles of any segment of a circle are equal) should, in earlier text-books, have ;^ :

been placed before the theorem of Eucl. i. 5. We can hardly suppose it to ^>
have been proved otherwise than by the superposition of the semicndes into

which the circle is divided by the diameter which bisects at right angles the _^ A

base of the segment; and no doubt the proof would be closely connected 1^ "*
that of Thales' other proposition that any diameter of a circle bisects it, which
must also (as Proclus indicates) have been proved by superposing one of the
two parts upon the other.

It is a natural inference fix>m the passage of Aristode that Euclid's proof of

aS4 BOOK I D- 5

I. 5 was his own, and it would thus appear that his innovations as regards
order of propositions and methods of proof began at the very threshold of the
subject

Proof without producing the sides.
In this proof, given by Proclus (pp. 248, a a — 349, 19), D and E are taken
on AB^ AC, instead of on AB, Atprodttad, so that AD, AEwn, equal The
method of proof is of course exactly like Euclid's, but it does not establish the
equality of the angles beyond the base as well.

Pappus' proof.

Proclus (pp. 249, 20—250, 12) says that Pappus proved the theorem in a
still shorter manner without the help of any construction whatever.

This very interesting proof is given as follows :

" Let ABC be an isosceles triangle, and AB equal to
AC.

Let us conceive this one triangle as two triangles, and let
us arfl;ue in this way.

Smce AB is equal to AC, and AC to AB,
the two sides AB, AC are equal to the two sides AC, AB.

And the angle BAC is equal to the angle CAB, for it is
the same.

Therefore all the corresponding parts (in the triangles) are equal, namely

BC to BC,

the triangle ABC to the triangle ABC (Le. ACB),

the angle ABC to the angle ACB,

and the angle ACB to the angle ABC,

(for these are the angles subtended by the equal sides AB, AC.

Therefore in isosceles triangles the angles at the base are equal."

This will no doubt be recognised as the foundation of the alternative
proof frequently given by modem editors, though they do not refer to Pappus.
But they state the proof in a different form, the common method being to
suppose the triangle to be taken up, turned over, and placed again upon stsetf,
after which the same considerations of congruence as those used by Euclid in
I. 4 are used over again. There is the obvious difficulty that it supposes the
triangle to be taken up and at the same time to remain where it is. (Cf.
Dodgson's humorous remark upon this, Euclid and his modem Rivals, p. 47.)
Whatever we may say in justification of the proceeding (e.g. that the triangle
may be supposed to leave a trace), it is really equi^ent to assuming &e
construction (hypothetical, if you will) of another triangle equal in all respects
to the given triangle ; and such an assumption is not in accordance with
Euclid's principles and practice.

It seems to me that the form given to the proof by Pappus himself is by &r
the best, for the reasons (i) that it assumes no construction of a second
triangle, real or hypothetiad, (3) that it avoids the distinct awkwardness
involved by a proof which, instead of merely quoting and applying the result
of a previous proposition, repeats, with reference to a new set of data, the
process by which that result was established. If it is asked how we are to
realise Pappus' idea of two triangles, surely we may answer that we keep to one
triangle and merely view it in two aspects. If it were a question of helping a
b^;inner to understand this, we might say that one triangle is the triangle

\

t

J

I

I. s, 6] PROPOSITIONS 5, 6 255

looked at in front and that the other triangle is the same triangle looked at
from behind] but even this is not really necessary.

Pappus' proof, of course, does not include the proof of the second part of
the proposition about the angles under the base, and we should still have to
establish this much in the same way as Euclid does.

Purpose of the second part of the theorem.

An interesting question arises as to the reason for Euclid's insertion of the
second part, to which, it will be observed, the converse proposition i. 6 has
nothing corresponding. As a matter of fact, it is not necessary for any
subsequent demonstration that is to be found in the original text of Euclid,
but only for the interpolated second case of i. 7; and it was perhaps not
unnatural that the undoubted genuineness of the second part of i. 5 convinced
many editors that the second case of i. 7 must necessarily be Euclid's also.
Proclus' explanation, which must apparently be the right one, is that the
second part of i. 5 was inserted for the purpose of fore-arming the learner
against a possible objection (Ivoroo-ic), as it was technically called, which might
be raised to i. 7 as given in the text, with one case only. The objection would,
as we have seen, take the specific ground that, as demonstrated, the theorem
was not conclusive, since it did not cover all possible cases. From this point
of view, the second part of l 5 is useful not only for i. 7 but, according to
Proclus, for i. 9 also. Simson does not seem to have grasped Proclus'
meaning, for he says : " And Proclus acknowledges, that the second part of
Prop. 5 was added upon account of Prop. 7 but gives a ridiculous reason for
it, 'that it might afford an answer to objections made against the 7th,' as if the
case of the 7th which is left out were, as he expressly makes it, an objection
against the proposition itself."

Proposition 6.

If in a triangle two angles be equal to one another^ the
sides which subtend the equal angles will also be equal to one
another.

Let ABC be a triangle having the angle ABC equal to
the angle ACB\

I say that the side AB is also equal to the
^ side AC.

j For, if AB is unequal to AC, one of them is

greater.

Let AB be greater; and from AB the
greater let DB be cut off equal to -^C the less ;

let DC be joined.

Then, since DB is equal to AC,
and BC is common,

the two sides DB, BC are equal to the two sides AC,
CB respectively ;

956 BOOK I [l6

and the angle DBC is equal to the angle ACB ;

therefore the base DC is equal to the base AB^
and the triangle DBC will be equal to the triangle ACB^

the less to the greater :
which is absurd

Therefore AB is not unequal to AC;
it is therefore equal to it
Therefore etc.

Q. E. D.

Euclid assumes that, because Z> is between A and B, the triangle DBC
13 less than the triangle ABC. Some postulate is necessary to justify diis
tadt assumption; considering an angle less than two right angles, say the
angle ACB in the figure of tiie pr(qx)sition, as a cluster df rays issuing fiom
C and bounded by the rays CA^ CB^ and joining AB (where A,B$xe any
two points on CA^ CB respectively), we see that to each successive ray taken
in the direction from CA to CB there corresponds one point on AB in whidi
the said ray intersects AB^ and that all the points on AB taken in <mler from
A to B correspond univocally to all the ra^ taken in order frcun CA to
CB, each point namely to the ray intersectmg AB in the point

We have here used, for the &rst time in the ElementSf the method of
reductio ad absurdum^ as to which I would refer to the section above (pp. 136,
140) dealing with this among other technical terms.

This proposition also, being the converse of the preceding proposition,
brings us to the subject of

Geometrical Conversion.

This must of course be distinguished from the logical conversion of a
proposition. Thus, from the proposition that all isosceles triangles have the
angles opposite to the equal sides ecjual, logical conversion would only enable
us to conclude that some triangles with two angles equal are isosceles. Thus
I. 6 is the geometrical, but not the logical, converse of i. 5. On the other
hand, as De Morgan points out {Companion to the Almanac, 1849, p. 7), L 6 is
a purely ^Ss^o/ deduction from i. 5 and i. 18 taken together, as is i. 19 aba
For the general argument see the note on i. 19. For die present proposition
it is enough to state the matter thus. Let X denote the class of triangles
which have the two sides other than the base equal, Y the class of triangles
which have the base angles equal \ then we may call non-A' the class of
triangles having the sides other than the base unequal, non- Y the class of
triangles having the base angles unequal

Thus we have

All AT is K, [i. s]

All lion-X is non-K; [i. 18]
and it is a purely logical deduction that

All y is AT. [i. 6]

According to Produs (p. 352, 5 sqq.) two forms of geometrical conversion
were distinguished.

(i) The leading form {jrpmfyoviUrq), the oonvemoTi par excellence (1I1 KVfimi

1

f

1.6] PROPOSITION 6 15?

ayruFTfio^, is the complete or simple conversion in which the hypothesis
and the conclusion of a theorem change places exactly, the conclusion of the
theorem bein^ the hypothesis of the converse theorem, which again establishes,
as its conclusion, the hypothesis of the original theorem. The relation between
the first part of i. 5 and i. 6 is of this character. In the former the hypothesis
is that two sides of a triangle are equal and the conclusion is that the angles
at the base are equal, while the converse (i. 6) starts from the hypothesis that
two angles are equal and proves that the sides subtending them are equal.

(2) The other form of conversion, which we may call partial^ is seen
in cases where a theorem starts from two or more hypotheses combined into
one enunciation and leads to a certain conclusion, liter which the converse
theorem takes this conclusion in substitution for one of the hypotheses of
the original theorem and from the said conclusion along with the rest of the
original hypotheses obtains, as its conclusion, the omitted hypothesis of the
original theorem, i. 8 is in this sense a converse proposition to i. 4 ; for i. 4
takes as hypotheses (i) that two sides in two triangles are respectively equal,
(3) that the included angles are equal, and proves (3) that the bases are equal,
while I. 8 takes (i) and (3) as hypotheses and proves (2) as its conclusion. It
is clear that a conversion of the leading type must be unique, while there
may be many partial conversions of a theorem according to the number of
hjrpotheses from which it starts.

Further, of convertible theorems, those which took as their hypothesis
the gtnus and proved a property were distinguished as the leading theorems
(irpoi;yov/tcva), while those which started from the property as hypothesis
and described, as the conclusion, the genus possessing that property were the
converse theorems, i. 5 is thus the leading theorem and i. 6 its converse,
since the genus is in this case taken to be the isosceles triangle.

Converse of second part of I. 5.

Why, asks Proclus, did not Euclid convert the second part of i. 5 as well ?
He suggests, properly enough, two reasons: (i) that the second part of i. 5
itself is not wanted for any proof occurring in the original text, but is only put
in to enable objections to the existing form of later propositions to be met,
whereas the converse is not even wanted for this purpose \ (2) that the converse
could be deduced from i. 6, if wanted, at any time after we have passed i. 13,
which can be used to prove that, if the angles formed by producing two sides
of a triangle beyond the base are equal, the base angles themselves are equal.

Proclus adds a proof of the converse of the second part of i. 5, Le. of the
proposition that, if the angles formed by producing two
sides of a triangle beyond the base are equal, the triangle
is isosceles; but it runs to some length and then only
effects a reduction to the theorem of i. 6 as we have it.
As the result of this should hardly be assumed, a better
proof would be an independent one adapting Euclid's
own method in i. 6. Thus, with the construction of i. 5,
we first prove by means of i. 4 that the triangles BFC^
CGB are equal in aU respects, and therefore that FC is
equal to GB^ and the angle BFC equal to the angle CGB.
Then we have to prove that AF^ AG sure equal. If they
are not, let AF be the greater, and from FA cut off FH equal to GA.
Join CH.

H. E. 17

JS8 hOOlt t ti.6,7

Then we have, in the two triaz^les HFC^ AGB^

two sides HF^ FC equal to two sides AG^ GB
and the angle ^/C equal to the angle AGB.

Therefore (i. 4) the triangles HFC^ AGB are equal But the triangles
BFC, CGB are also equal

Therefore (if we take away these equals respectively) the triangles HBQ
ACB are equal: which is impossible.

Therefore AF^ AG are not unequal.

Hence AF\& equal to AG and, if we subtract the equals BF^ CG respec-
tively, AB is equal to AC.

This proof is found in the commentary of an-Naiibl (ed. Besthom-Heibeigi
p. 61 ; ed. Curtze, p. 50).

Alternative proofs of I. 6.

Todhunter points out that i. 6, not being wanted till 11. 4, could be
postponed till later and proved by means of i. 26. Bisect the angle BAC
by a straight line meeting the base at 2>. Then die triangles ABD^ ACD
are equal in all respects.

Another method depending on i« 26 is given by an-Nairld after that
proposition.

Measure equal lengths BD^ CE along the sides BA^ CA.
Join BE, CD.

Then [i. 4] the triangles DBC, ECB are equal in all
respects;

therefore £B, DC are equal, and the angles BEC, CDB
are equal.

The supplements of the latter angles are equal [1. 13],
and hence the triangles ABE, ACD luive two angles equal respectively and
the side BE equal to the side CD.

Therefore [1. 26] AB is equal to AC.

Proposition 7.

Given two straight, lines constructed on a straight line
{from its extremities) and fneeting in a point, there cannot be
constructed on the same straight line {from its extremities\
and on the same side of it, two other straight lines meeting tn

5 another point and equal to the former two respectively, namely
each to that which has the same extremity with it.

For, if possible, given t^o straight lines AC, CB con-
structed on the straight line AB and meeting
at the point C, let two other straight lines

10 AD, DB be constructed on the same straight
line AB, on the same side of it, meeting in
another point D and equal to the former two
respectively, namely each to that which has
the same extremity with it, so that CA is

15 equal to DA which has the same extremity A with it, and

,1

L 7] PROPOSITIONS 6, 7 ^59

CB to DB which has the same extremity B with it ; and let
CD be joined.

Then, since -^C is equal to ADy

the angle ACD is also equal to the angle ADC; [i. s]

therefore the angle ADC is greater than the angle DCB ;

therefore the angle CDB is much greater than the ancfle
DCB.
\ Again, since CB is equal to DB,
" \ the angle CDB is also equal to the angle DCB.

■ |; But it was also proved much greater than it :

which is impossible.
Therefore etc. q. e. d.

i!

1—6. In an English translation of the enunciation of this proposition it is absolutely
"...... . wnicn . . -

I The reason is partly that the Greek enunciation is itself very elliptioU, and partly that some
words used in it conveyed more meaning than the corresponding words in English do.
Particularly is this the case with 06 ffvera^orrai M "there shall not be constructed upon,"
since ^wi^rmff9ai is the regular word for constructing a triangle in particular. Thus a Ureek

' would easily understand vwraBi/faoirrai 4vl as meaning the construction of two Unts forming'
a triangle on a given straight line as base ; whereas to "construct two straight lines on a
straight line*' is not in English sufficiently de6nite unless we explain that they are drawn
from the ends of the straight line to meet at a point. I have had the less hesitation in putting
in the words "from its extremities" because they are actually used by Euclid in the somewhat
similar enunciation of I. 91.

How impossible a literal translation into English is, if it is to convey the meaning of the
enunciation mtelligibly, will be clear from the following attempt to render literally: **On the
same straight line there shall not be constructed two other straight lines equal, each to each,
to the same two straight lines, (terminatixig) at different points on the same side, having the
same extremities as the original straight lines *' (M r^f 0^1% tMtlat M rsut adroit eiStUut

\tidp^ tA oMl Wpara ix^wm rtus ^ d^X^t ci^^eiait).

i The reason whv Euclid allowed himself to use, in this enunciation, language apparently

I so obscure is no doubt that the phraseology was traditional and therefore, vague as it was,

I had a conventional meaning whidi the contemporary geometer well understood. This is

I proved, I think, by the occurrence in Aristotle {Metearvlogiea lii. 5, 376 a 2 sqq.) of the very

'same, evidently technical, expressions. Aristotle is there alluding to the theorem given by

j Entodus from Apollonius' Plane Loci to the effect that, if /T, ^ be two fixed points and M

such a variable point that the ratio of MH to MK is a given ratio (not one of equality^ the

locus of A/ is a circle. (For an account of this theorem see note on vi. 3 below.) Now

Aristotle says " The lines drawn up from /^, AT in this ratio cannot be constructed to two

different points of the semicircle ^ ** (ol oj^r drd rQnf HK dMiY^/««yai ypoLtiftaX ^ rodry r^

Xiytf oi ovvTaSiiVQwrai roO i^l', f A iifUKWcktov vpbt AWo Kal dXXo ffii/ittow).

If a paraphrase is allowed instead of a translation adhering as closely as possible to the
original, Simson's is the best that could be found, since the fiiict that the straight lines form
triimgles on the same base is really conveyed in the Greek. Simson*s enunciation is. Upon
the same Aau, and on the same side of it^ there cannot be two triangles that have their sides
which are terminated in one extremity of the base equal to one another^ and likewise those
vihieh art terminated at the other extremity, Th. Taylor (the translator of Produs) attacks
'imson's alteration as "indiscreet" and as detractmg from the beauty and accuracy of
Jodid's enunciation which are enlarged upon by Proclus in his commentary. Yet, when
Taylor says ** Whatever difficulty learners may find in conceiving this proposition abstractedly
b CMily removed by its exposition in the figure," he really gives his case away. The fact is
l|that Taylor, always enthusiastic over his author,, was nettled by Simson's sighting remarks
on Proclus' comments on the proposition. Simson had said, with reference to Proclus'
explanation of the bearing of the second part of i. 5 on i. 7, that it was not "worth while

17 — 2

36o fiook t [1.7

to relate his trifles mt full length," to whidi Taylor retorts ''But Mr Simaoo was no
philosopher ; and therefore the greatest part of these Commentaries must be oonsiderad fay
nim as trifles, from the want of a phuoaophic genius to comprehend their meaning, and
a taste superior to that of a nc/rv mmthmtUidam^ to discover their beauty and d^guoe.**

so. It would be natural to insert here the step '*lmt the angle ACD is greater than the
angle ^C/>. [C. A^. 5.]"

1 1 . much greater, literally " greater by much " (voXXf /mI^) • Simsoo and thoae who
follow him transUte : ^^much mwn tkm is the angfle BDC grmitr than the anrie BCD^^
but the Greek for this would have tobe veXX^ (or a-oXv) ^kWkw ^#n...#Mf^. nXXf fJOJm^
however, though used by Apollonius, Is not, apparently, found in Endid or Archimedes.

Just as in 1. 6 we need a Postulate to justify tbeoreticaQy die statement that
CD falls within the angle ACB^ so that the triangle DBC is less than the
triangle ABQ so here we need Postulates which shall satisfy us as to the
relative positions of CA^ CB^ CD on the one hand and of />C, DA^ DB
on the other, in order that we may be able to infer that the angle BDC is
greater than the angle ADC^ and the angle ACD greater than the angle BCD.

De Morgan {op. at. p. 7) observes that 1. 7 would be made easy to
beginners if they were first familiarised, as a common notion, with **if two
magnitudes be equal, any magnitude greater than the one is greater than any
magnitude less than the other." I doubt however whether a beginner would
follow this easily ; perhaps it would be more easily apprehended in the form
''if any magnitude A is greater than a magnitude B^ the magnitude A is
(a fortiori) greater than any magnitude less than B or than any magnitudie
equal to B."

It has been mentioned already (note on i. 5) that the second case of l 7
given by Simson and in our text-books generally is not in the original text
(the omission being in accordance with Euclid's general practice of giving
only one case, and tiiat the most difficult, and leaving the others to be worked
out by the reader for himself). The second case is given by Produs as the
answer to a possible objection to Euclid's proposition, which should assert that
the proposition is not proved to be universally true, since the proof given does
not cover all possible cases. Here the objector is supposed to contend that
what Euclid declares to be impossible may still be possible if one pair of lines
lie wholly within the other pair of lines; and the second part of i. 5 enables
the objection to be refuted.

If possible, let AD^ DB be entirely within the triangle formed by AC^
CB with ABy and let AC be equal to AD and BC \

to BD.

Join CD^ and produce AC^ AD to E and F.

Then, since AC is equal to AD^

the triangle ACD is isosceles,

and the angles ECD^ FDC under the base are equal.
But the angle ECD is greater than the angle BCD \

therefore the angle FDC is also greater than the angle

BCD.

Therefore the angle BDC is greater by far than the angle BCD.

Again, since DB is equal to CB^
the angles at the base of the triangle BDC are equal, [i. 5 ^

that is, the angle BDC is equal to the angle BCD.

Therefore the same angle BDC is both greater than and equal to the ai^le
BCD: which is impossible.

The case in which D CeJIs on AC or BC does not require proof.

i

I. 7, 8] PROPOSITIONS 7, 8 261

I have already referred (note on i. i) to the mistake made by those
editors who regard i. ^ as being of no use except to prove i. 8. What i. 7
proves is that if[ in addition to the base of a triangle, the length of the side
terminating at each extremity of the base is given, only one triangle satisfying
these conditions can be constructed on one and the same side of the given
base. Hence not only does i. 7 enable us to prove i. 8, but it supplements
1. I and I. 22 by showing that the constructions of those propositions ^ve one
triangle only on one and the same side of the base. But for i. 7 this could
not be proved except by anticipating iii. 10, of which therefore i. 7 is the
equivalent for Book i. purposes. Dodgson (Euclid and his modem Rivals^
pp. 194 — 5) puts it in another way. " It [i. 7] shows that, of all plane figures
that can be made by hingeing rods together, the thrte-siAeA ones (and these
only) are rigid (which is another way of stating the fact that there cannot be
iw0 such figures on the same base). This is imalogous to the fact, in relation
to solids contained by plane surfaces hinged together, that any such solid is
rigid, there being no maximum number of sides. And there is a close analogy
between i. 7, 8 and iii. 23, 24. These analogies give to geometry much of its
beauty, and I think that they ousht not to be lost sight of." It will therefore
be apparent how ill-advised are those editors who eliminate i. 7 altogether and
rely on Philo's proof for i. 8.

• Proclus, it may be added, gives (pp. 268, 19 — 269, 10) another explanation
of the retention of i. 7, notwithstanding that it was apparently only required
for I. 8. It was said that astronomers used it to prove that three successive
eclipses could not occur at equal intervals of time, i.e. that the third could not
follow the second at the same interval as the second followed the first ; and it
was ai|;ued that Euclid had an eye to this astronomical application of the
proposition. But, as we have seen, there are other grounds for retaining the
proposition which are quite sufficient of themselves.

Proposition 8.

1/ two triangles have the two sides equal to two sides

respectively, and have also the base equal to t/ie base, they will

aba have the angles equal which are contained by the equal

straight lines.

\ Let ABC, DEF be two triangles having the two sides

\AB, AC equal to the two sides

^DE, DF respectively, namely

.AB to DE, and AC to DF\ and

^let them have the base BC equal

to the base EF\

1 say that the angle BAC is
also equal to the angle EDF.

For, if the triangle ABC be
applied to the triangle DEF, and if the point B be placed on
[the point E and the straight line BC on EF,
the point C will also coincide with F,
because BC is equal to EF.

263 BOOK I [l8

Then, BC coinciding with EF^
BA. AC will also coincide with ED, DF\
» for, if the base BC coincides with the base EF, and the sides
BA, AC do not coincide with ED, DF but fall beside them
as EG, GF,

then, given two straight lines constructed on a straight

line (from its extremities) and meeting in a point, there will

sshave been constructed on the same straight line (from its

extremities^, and on the same side of it, two other straight

lines meeting in another point and equal to the former

two respectively, namely each to that which has the same

extremity with it

30 But they cannot be so constructed. [i. 7]

Therefore it is not possible that, if the base BC be applied

to the base EF, the sides BA, AC should not coincide with

ED, DF;

they will therefore coincide,
35 so that the angle BAC will also coincide with the angle
EDF, and will be equal to it

If therefore etc. q. e. d.

19. B A, AC. The text has here " BA, CA, "

91. fall beside them. The Greek has the future, wa^taXk^qvvi, wapoXKirrm means
" to pass by without touchiug,** ** to miss" or *' to deviate."

As pointed out above (p. 257) i. 8 is a /or/Za/ converse of i. 4.

It is to be observed that in i. 8 Euclid is satisfied with proving the equality
of the vertical angles and does not, as in i. 4, add that the triangles are equal,
and the remaining angles are equal respectively. The reason is no doubt (as
pointed out by Proclus and by Savile afler him) that, when once the vertical
angles are proved equal, the rest follows from i. 4, and there is no object in
proving again what has been proved already.

Anstotle has an allusion to the theorem of this proposition in MeteorologUa
III. 3, 373 a 5 — 16. He is speaking of the rainbow and observes that, if equal
ra3rs be reflected from one and the same point to one and the same point, the
points at which reflection takes place are on the circumference of a circle.
"For let the broken lines ACB, AFB, ADB be all reflected from the point
A to the point B (in such a way that) AC, AF, AD are all equal to one
another, and the lines (terminating) at B, Le. CB, FB, DB, are likewise all
equal ; and let AEB be joined. It follows that the triangles are equal; for
they are upon the equal (base) AEB*^

Heibe^ {Mathematisches nt Aristoteles, p. 18) thinks that the form of the
conclusion quoted is an indication that in the corresponding proposition to
Eud. I. 8, as it lay before Aristotle, it was maintained that the triangles were
equal, and not only the angles, and "we see here therefore, in a clear example,
how the stones of the ancient fabric were recut for the rigid structure of his

18]

PROPOSITION 8

363

I

I

Elements.^ I do not, however, think that this inference from Aristotle's
language as to the form of the pre-Euclidean proposition is safe. Thus if we,
nowadays, were arguing from the data in the passage of Aristotle, we should
doubtless infer dir^y that the triangles are equal in all respects, quoting i. 8
alone. Besides, Aristotle's language is rather careless, as the next sentences
of the same passage show. " Let perpendiculars,"
he says, *'be drawn to AEB from the angles, CE
from C, FE from ^and DE from D. These, then,
are equal j for they are all in equal triangles, and
in one plane; for all of them are perpendicular
to AEB^ and they meet at one point E. There-
fore the (line) drawn (through C, F, 2P) will be a
circle, and its centre (will be) E'* Aristotle should
obviously have proved that the three perpendiculars will meet at one point E
on AEB before he spoke of drawing the perpendiculars CE^ FE, DE,
This of course follows from their being " in equal triangles " (by means of
EucL I. 26); and then, from the fact that the perpendiculars meet at one
point on AB, it can be inferred that all three are m one plane.

Philo'8 proof of I. 8.

This alternative proof avoids the use of i. 7, and it is elegant ; but it is
inconvenient in one respect, since three cases have to be distinguished.
Proclus gives the proof in the following order (pp. 266, 15 — 268, 14).

I^ ABC, DEF be two triangles having the sides A By -<4C equal to the
sides DEy 2?^ respectively, and the base ^C equal to the base EF,

Let the triangle ABC be applied to the triangle DEF, so that B is placed
on E and BC on EF, but so that A falls on the opposite side of EF from D,
taking the position G. Then C will coincide with F, since BC is equal to
EF.

Now FG will either be in a straight line with DF, or make an angle with
it, and in the latter case the angle will either be interior (icara to ivro^) to the
figure or exterior (icara ro licw).

I. Let FG he in a straight line with
DF

Then, since DE is equal to EG, and
DFG is a straight line,

DEG is an isosceles triangle, and the
angle at Z^ is equal to the angle at G.

[I- s].

II. Let DFy FG form an angle interior to the figure
Let DG be joined.
Then, since DE, EG are equal,

the angle EDG is equal to the angle
EGD.

Again, since DF\& equal to FG,
the angle FDG is equal to the angle
FGD.

Therefore, by addition,
the whole angle EDF is equal to the
whole angle EGF,

364 BOOK I [1.8,

n

III. Let DF^ FG form an angle txterwr to the figure.

Let DG be joined.

The proof poceeds as in the last case,
except that subtraction takes the place of
addition, and

the remaining angle EDF is equal to the
remaining angle EOF.

Therefore in all three cases the angle
EDF is equal to the angle EGF^ that is,
to the angle >ff^C.

It will be observed that, in accordance widi the practice of the Greek
geometers in not recognising as an "angle " aiw angle not less than two right
angles, the re-entrant angle of the quadrilateral JDEuF\% ignored and the ang^
DFG is said to be outside the figure.

Proposition 9.

To bisect a given rectilineal angle.

Let the angle BAC be the ^ven rectilineal angle.

It is then required to bisect it.

Let a point D be taken at random on AB ;
let AE be cut off from AC equal to AD ; [i. 3]
let DE be joined, and on DE let the equilateral
triangle DEFh^ constructed ;
let .^/^ be joined.

I say that the angle BAC has been bisected by the
straight line AF.

r or, since AD is equal to AE,
and AF is common,

the two sides DA, AF are equal to the two sides ^
EA, AF respectively.

And the base DF is equal to the base EF\

therefore the angle DAF is equal to the angle EAF. j

[I. 8] I

Therefore the given rectilineal angle BAC has been |

bisected by the straight line AF. Q. e. f. ^

It win be observed from the translation of this proposition that Euclid
does not say, in his description of the construction, that the equilateral triangle |
should be constructed on the side of DE opposite to ^ ; he leaves this to be <
inferred from his figure. There is no particular value in Proclus' explanation \
as to how we should proceed in case any one should assert that he could not .
recognise the existence of any space below DE. He supposes, then, the !
equilateral triangle described on the side of DE towards A, and hence has to -.
consider three cases according as the vertex of the equilateral triangle fidls
on A, above A or below it The second and third cases do not difier

i

I. 9] PROPOSITIONS 8, 9 265

substantially from Euclid's. In the first case, where ADE is the equilateral
triangle constructed on DE^ take any point ^on AD^ and from AE cut off
AG equal to AF, Join DG^ J?^ meeting in H\ and
join AH, Then ^^ is the bisector required.

Proclus ako answers the possible AjecHon that
might be raised to Euclid's proof on the ground that
it assumes that, if the equilateral triangle m described
on the side of DE opposite to A^ its vertex -F will lie
within the angle BAC. The objector is supposed to
argue that this is not necessary, but that ^ might fall
eiSier on one of the lines forming the angle or outside
it altogether. The two cases are disposed of thus.

Suppose ^to fall as shown in the two figures below respectively.

Then, since FD is equal to FE^
the angle FDE is equal to the angle FED,

Therefore the angle CED is greater than the angle FDE ; and, in the
second figure, a fortiori^ the angle CED is greater than the angle BDE.

But, since ADE is an isosceles triangle, and the equal sides are produced.

the angles under the base are equal,

i.e., the angle CED is equal to the angle BDE,

But the an^e CED was proved greater : which is impossible.

Here then is the second case in which, in Proclus' view, the second part
of I. 5 is useful for refuting objections.

On this proposition Proclus takes occasion (p. 371, 15—19) to emphasize
I the isjct that the given angle must be rectilineal^ since the bisection of any sort

of angle (including angles made by curves with one another or with straight
\ lines) is not matter for an elementary treatise, besides which it is questionable

^whether such bisection is always possible. "Thus it is difficult to say

iwhether it is possible to bisect the so-called horn-like angle " (formed by the

prcumference of a circle and a tangent to it).

Trisection of an angle.

Further it is here that Proclus gives us his valuable historical note about
ithe trisection of any acute angle, which (as well as the division of an angle in
iny ^ven ratio) requires resort to other curves than circles, i.e. curves of the
speaes which, after Geminus, he calls "mixed." "This," he says (p. 372,
I — 12), "is shown by those who have set themselves the task of trisecting such
a given rectilineal angle. For Nicomedes trisected any rectilineal anple by
means of the conchoidal lines, the origin, order, and properties of iduch he
has handed down to us, being himself the discoverer of their peculiarity.
Others have done the same thing by means of the quadratrices of Hippias
..and Nicomedes, thereby again using 'mixed' curves. ^ But others, starting
I' from the Archimedean spirals, cut a given rectilineal angle in a given ratio."

* ■ r

366

-JLJLl "•"■■>

BOOK I

[1.9

(a) Trisection by means of the amckaid.

I have already spoken of the ionchmd of Nioomedes (nole on Det a,
pp. 160 — i) ; it remains to show how it could be nsed for trisecting an
angle. Pappus explains this (iv. pp. 374 — 5) as follows.

Let ABC be the given acute angle, and from any point A in AB draw
A C perpendicular to BC.

e O

Complete the parallelogram FBCA and produce FA \xi z, point E luch
that, if BE be joined, BE interapis between AC and AE a Ungtk DE fual
to twice AB.

I say that the angle BBC is one-third of the angle ABC

For, joining A to G, the middle point of DE^ we have the three straUht
. lines AG, DG, EG equal, and the angle AGD is double of the angle AED
or EBC

But DE is double of AB ;
therefore AG, which is equal to DG, is equal to AB.

Hence the angle AGD is equal to the angle ABG.

Therefore the angle ABD is also double of the angle EBC;
so that the angle EBC is one-third of the angle ABC,

So bi Pappus, who reduces the construction to the drawing of BE so
that DE shall be equal to twice AB.

This is what the conchoid constructed with Bz&pole, AC2& directrix^ and
distance equal to twice AB enables us to do ; for that conchoid cuts AE in
the required point E.

{b) Use of the quadratrix.

The plural quadratrices in the above passage is a Hellenism for the
singular quadratrix, which was a curve discovered by Hippias of Elis about
4^ B.C. According to Produs (p. 356, 11) Hippias proved its properties ;
and we are told (i) in the passage quoted above that Nicomedes also
investi^ted it and Uiat it was used for trisecdi^ an angle, and (3) by Pappus
(iv. pp. 250, 33 — 252, 4) that it was used by Dmostratus and Nicomedes and
some more recent writers for squaring the circle, whence its name. It is
described thus (Pappus iv. p. 252).

Suppose that ABCD is a square and BED a quadrant of a circle with
centred.

Suppose (i) that a radius of the circle moves
uniformly about A from the position AB to the
position AD, and (2) that in the same time the
line BC moves uniformly, always parallel to itself,
and with its jextremity B moviiig along BA, from
the position BC to the position AD.

Then the radius AE and the moving line BC
determine at any instant by their intersection a
point F.

The locus of ^is the quadratrix.

p

l

'^

V ♦

1 1

!iQ i

>

, I. 9, lo] PROPOSITIONS 9, 10 267

1 The property of the curve is that, if ^ is any point, the arc BED is
• to the arc ED as AB is to FH.

i In other words, if ^ is the angle FAD^ p the radius vector AFand a the
aide of the square,

(p8in^)/a = ^/Jir.

^ Now the angle EAD can not only be trisected but divided in any git*en

I ratio by means of the quadratrix (Pappus iv. p. 286).

I Fof let FH be divided at K in the given ratio.

I Draw KL parallel to AD^ meeting the curve in L ; join AL and produce

I it to meet the circle in N.

Then the angles EAN^ NAD are in the ratio of FK to KH^ as is easily
proved.

(c) Use of the spiral of Archimedes.

The trisection of an angle, or the division of an angle in any ratio, by
means of the spiral of Archimedes is of course an equally simple matter.
Suppose any angle included between the two radii vectores OA and OB of the
^^Md, and let it be required to cut the angle AOB in a given ratio. Since
the radius vector increases proportionally with the angle described by the
sector which generates the curve (reckoned from the original position of the
(rector coinciding with the initial line to the particular position assiuned), we
iiave only to take the radius vector OB (the greater of the two OA^ 0B\
nark off OC along it equal to OA^ cut CB in the given ratio (at D say), and
fhen draw the circle with centre O and radius OD cutting the spiral in E,
uhen OE will divide the angle AOB in the required manner.

I Proposition id.

To bisect a given finite straight line.

Let AB be the given finite straight line.
Thus It is required to bisect the finite straight line AB.
Let the equilateral triangle ABC be
.K>nstructed on it, [i. i]

ind let the angle ACB be bisected by the
traight line CD ; [1. 9]

\ I say that the straight line AB has
■•en bisected at the point D.
- For, since ACv& equal to CB,
id CD is common,

I the two sides AC, CD are equal to the two sides Bd
'Z? respectively ;

nd the angle A CD is equal to the angle BCD ;

\ therefore the base AD is equal to the base BD. [i. 4}

iTherefore the given finite straight line AB has been
lisected at D. q. e. f.

268

BOOK I

[I. lol

[1.8;

Apollonius, we are told (Produs, pp. 279, 16— 280, 4), bisected a stimight
line AB by a construction like that of i. i.
With centres A^ B, and raiUi AB^ BA respec-
tively, two circles are described, intersecting in
C, D. ]o\nmg CD, AC, CB, AD, DB, Apol-
lonius proves in two steps that CD bisects AB.

(i) Since, in the triangles A CD, BCD,
two sides AC, CD are equal to two sides
BC, CD,
and the bases AD, BD are equal,
the angle A CD is equal to the angle
BCD.

(2) The latter angles being equal, and AC being equal to CB^ while C£
is common,

the equality of AE, EB follows by l 4.

The objection to this proof is that, instead of assuming the bisection o
the angle ACB, as already effected by i. ^ ApoUonius goes a step fiuthe»
back and embodies a construction for bisecting the ang^ That iS| bt
unnecessarily does over again what has been done before, which is open to
objection from a theoretiad point of view.

Proclus (pp. 277, 25 — 279, 4) warns us against being moved by thisi
proposition to conclude that geometers assumed, as a preliminary hypothesis. |
that a line is not made up of indivisible parts (If Vip«r). This might bt|
argued thus. If a line is made up of indivisibles, there must be in a finitti
line either an odd or an even number of them. If the number were odd,!
it would be necessary in order to bisect the line to bisect an indivisible (the)
odd one). In that case therefore it would not be possible to bisect a straight
line, if it is a magnitude made up of indivisibles. But, if it is not so made
up, the straight line can be divided ad infinitum or without limit (hr cnrcipor
Staipciroi). Hence it was argued (^oo-u'), says Proclus, that the divisibility
of magnitudes without limit was admitted and assumed as a geometrical
principle. To this he replies, following (jeminus, that geometers did indeeci
assume, by way of a common notion, that a continuous magnitude, Le.
magnitude consisting of parts connected together (ounififMHtfr), is diviribli
(8uup«Tov). But infinite divisibility was not assumed by them ; it was pravei
by means of the first principles applicable to the case. **For when," h
sa3rs, ''they prove that the incommensurable exists among magnitudes, an
that it is not all things that are commensurable with one another, wlu
else will any one say that they prove but that every ma^tude can \
divided for ever, and that we sludl never arrive at the mdivisible, tb*
is, the least common measiure of the magnitudes? This then is matto* .
demonstration, whereas it is an axiom that everything continuous is divisible
so that a finite continuous line is divisible. The writer of the Elemeni
bisects a finite straight line, starting from the latter notion, and not fix»m an
assumption that it is divisible without limit ** Proclus adds that the propositio
may also serve to refute Xenocrates' theory of indivisible lines (aro^ ypofifjuai
The aigument given by Proclus to disprove the existence of indivisible linr
is substantially that used by Aristotle as r^;ards magnitudes generally (c
PAysics VI. I, 231 a 21 sqq. and especially vi. 2, 233 b 15 — 32).

Ill] PROPOSITIONS lo, II 269

Proposition ii.

I To draw a straight line at right angles to a given straight

line front a given point on it.

Let AB be the given straight line, and C the given point
on it.

Thus it is required to draw from the point C a straight
. line at right angles to the straight
\ line AB.

I Let a point D be taken at ran-
\ dom on AC\

^ let CE be made equal to CD ; [i. 3]
► on DE let the equilateral triangle
\FDE be constructed, • [i. i]
I and let FC be joined ;

I say that the straight line FC has been drawn at right
I angles to the given straight line AB from C the given point
Jon it.

• For, since DC is equal to CE,
rCF is common,
the two sides DC, CF are equal to the two sides EC,
I ' CF respectively ;

I and the base DF is equal to the base FE ;

therefore the angle DCF is equal to the angle ECF\

['• «]
and they are adjacent angles.

But, when a straight line set up on a straight line makes

the adjacent angles equal to one another, each of the equal

ingles is right ; [Def. 10]

therefore each of the angles DCF^ FCE is right

Therefore the straight line CF has been drawn at right

ngles to the given straight line AB from the given point

r on it.

Q. E. F.

10. let CE be made equal to CD. The verb is ntlrSw which, as well as the other
barts of ccifuu, is constantly used for the passive of rWriiu " to fiiace^ ; and the latter word
is constantly used in the sense of making, e.g., one straight line equal to another straight line.

De Morgan remarks that this proposition, which is " to bisect the angle
inade by a straight line and its continuation " [Le. a flat angle], should be a
particular case of i. 9, the constructions being the same. Tms is certainly

270

BOOK I

[i. lit IS

worth noting, though I doubt the advantage of rearranging the propositions
in consequence.

Apollonius gave a construction for this proposition (see Proclus, p. 282, 8)
differing from Euclid's in much the same way as his construction for bisecting
a straight line differed from that of i. 10. Instead of assuming an equilatenl
triangle drawn without repeating the process of i. i, Apollonius takes D and
E equidistant from C as in Euclid, and then draws circles in the manner of

I. I meeting at F. This necessitates proving again that DFv^ equal to FE\
whereas Euclid's assumption of the construction of i. i in the wc^ *' let the
equilateral triangle FDE be constructed " enables him to dispense with the
drawing of circles and with the proof that DF is equal to FE at the same
time. While however the substitution of Apollonius* constructions for l 10
and II would show faulty arrangement in a theoretical treatise like Eudid's,
they are entirely suitable for what we call practical geometry, and such may
have been Apollonius' object in these constructions and in his alternative for
I. 23.

Proclus f^ves a construction for drawing a straight line at right angles to
another straight line but from one end of it, instead of from an intermediate
point on it, it being supposed (for the sake of argument) that we are not
permitted to produce the straight line In the commentary of an-NaiiM (ed.
Besthom-Heiberg, pp. 73 — 4; ed. Curtze, pp. 54 — 5) this constructiv
attributed to Heron.

Let it be required to draw from A a straight line at right angles to AB. .

On AB take any point C, and in the manner of the proposition draw CE^
at right angles to AB,

From CE cut off CD equal to AC^ bisect the
angle ACE by the straight line CF^ [i. 9]

and draw DF at right angles to CE meeting CF
in F Join FA.

Then the angle FAC will be a right angle.

For, since, in the triangles ACF^ DCF^ the
two sides AC^ CF are equal to the two sides
DC^ CF respectively, and the included angles
ACF^ DCFzx^ equal,

the triangles are equal in all respects. [i. 4

Therefore the angle at A is equal to the angle at Z>, and is accordingly '
right angle.

E
F

D

\

A <

\ 6

Proposition 12.

To a given infinite straight line, from a given point
which is not on it, to draw a perpendicular straight line.

Let AB be the given infinite straight line, and C the
given point which is not on it ;

t

I

I. 12] PROPOSITIONS II, 12 271

5 thus it is required to draw to the given infinite straight
line AB^ from the given point
C which is not on it, a per-
pendicular straight line.

For let a point D be taken

10 at random on the other side of

the straight line AB, and with

centre c and distance CD let

the circle EFG be described ;

[Post 3]
let the straight line EG
IS be bisected at H, [i. 10]

and let the straight lines CG, CH, CE be joined.

[Post l]

I say that CH has been drawn perpendicular to the given
infinite straight line AB from the given point C which is
not on it.
20 For, since GH is equal to HE^
and HC is common,

flie two sides GH, HC are equal to the two sides
EH, HC respectively ;
and the base CG is equal to the base CE ;
25 therefore the angle CHG is equal to the angle EHC.

[1.8]

And they are adjacent angles.

But, when a straight line set up on a straight line makes

the adjacent angles equal to one another, each of the equal

angles is right, and the straight line standing on the other is

30 called a perpendicular to that on which it stands. [Def. 10]

Therefore C/^ has been drawn perpendicular to the given

infinite straight line AB from the given point C which is

not on it

Q. E. F.

3. a perpendicular straight line, KdBrrm edtfciov ypaf»M^p, This is the full expression
-for a perpiHdicular, ttiBrrw meaning lei faU or let dawn^ so that the expression corresponds
to our plumb-line, ^ jrdtfrrot b however constantly used alone for a perpendicular, ypofifiilj
being understood.

ID. on the other side of the straight line AB, literally '* towards the other parts of
the straight line AB,'' iwl rd Irfpa fU/ni r^t AB. Cf. **on the same side" (M rd o^d
ftdfni) in Post. 5 and **in both directions" (^* ixdnpa rd fU/ni) in Def. 33.

** This problem,*' says Produs (p. 283, 7—10), "was first investigated
by Oenopides [5th cent b.c], who thought it useful for astronomy. He
however calls the perpendicular, in the archaic manner^ (a line drawn)

272 BOOK I [i. 12

gmmum-wise (mra yr«fuyvaX because the gnomon is also at right anf^es to the
horizon." In this earlier sense the gnomon was a staff {daoed in a vertical
position for the purpose of casting shadows and so serving as a means of
measuring time (Cantor, GesckickU der Mathimatik^ i,, p. 161). The later
meanings of the word as used in Eucl. Book 11. and elsewhere will Le
explained in the note on Book lu Def. 2.

Proclus says that two kinds of perpendicular were distinguished^ the ''plane"
(IrtircSof) and the "solid" (orfpco), the former being the perpendicular
dropped on a line in a plane and the latter theperpenmcular dropped on a
plane. The term "solid perpendicular" is sufficiently curious, but it may
perhaps be compared with the Greek term ''solid locus" applied to a conic
section, apparently on the ground that it has its origin in the section of a
solid, namely a cone.

Attention is called by most editors to the assumption in this proposition
that, if only D be taken on the side oi AB remote from C, the circle described
with CD as radius must necessarily cut A£ in two points. To sati^ us of
this we need, as in i. i, some postulate of continuity, e.^. something like that
suggested by Killing (see note on the Principle of Contmuity above, pu 235}:
" If a point [here the point describing the arde] moves in a figure which is
divided into two parts [by the straight line], and if it belongs at the beginning
of the motion to one part and at another stage of the motion to the other
ps^ it must during the motion cut the boundary between the two parts," and
this of course applies to the motion in iufo directions fix>m D.

But the editors have not, as a rul^ noticed a possible ctjecHon to the
Euclidean statement of this problem which is much more difficult to dispose
of at this stage, i.e. without employing any proposition later than this in
Euclid's order. How do we know, says the supposed critic, that the cirde
does not cut AB in thra or more points, in which case there would be not
one perpendicular but three or more? Proclus (pp. 286, 12—289, 6) tries to
refute this objection, and it is interesting to follow his argument, though it
will easily be seen to be inconclusive. He takes in order three possible
suppositions.

I. May not the circle meet AB in a third point K between the middle
point of GE and either extremity of it, taking the form drawn in the figure
appended ?

Suppose this possible. Bisect GE in H. Join CJ7, and produce it to
• meet the circle in Z. Join CG^ CK, CE.

Then, since CG is equal to CE^ and
CH is common, while the base GH is
equal to the base HE^

the angles CHG^ CHE are equal and,
since they are adjacent, they are both right

Again, since CG is equal to CE^
the angles at G and E are equal.

Lastly, since CK is equal to CG and
also to CE, the angles CGK, CKG are
equal, as also are the angles CKE, CEK.

Since the angles CGK, CEJCare equal, it follows that

the angles CJCG, CITE are equal and therefore both right.
Therefore the angle CKHis equal to the angle CHK,
and CH\& equal to CK.

I. 12] PROPOSITION 12 273

But CK is equal to CZ, by the definition of the circle ; therefore CH is
equal to CL : which is impossible.

Thus Proclus; but why should not the circle meet AB m H2s well as JT?

2. May not the circle meet AB in H the middle point of GE and take
the form shown in the second fi^re?

In that case, says Proclus, jom CG^ CHy CE as before. Then bisect HE
at iT, join CK and produce it to meet
the circumference at 2.

Now, since HK is equal to KE^ CK
is common, and the base CH is equal to
the base CE^

the angles at K are equal and therefore
both right angles.

Therefore the angle CHK is equal to
the angle CKH^ whence CKv& equal to CH
and therefore to CL\ which is impossible.

So Proclus ; but why should not the circle meet AB in JT as well as /T?

3. May not the circle meet AB in two points besides G^ E and pass,
between those two points, to the side of AB towards C, as in the next figure ?

Here again, b^ the same method, Proclus proves that, K^ L being the
other two points m which the circle cuts
AB,

CK is equal to CH,
and, since the circle cuts CH in M,

CM is equal to CK and therefore to

tCH\ which is impossible.
But, again, why should the circle not
\ cut AB in the point Hz& well?

In &ct, Proclus' cases are not mutually
exclusive, and his method of proof only enables us to show that, if the circle
meets AB in one more point besides G^ E^ it must meet it in more points
still We can always find a new point of intersection by bisecting the distance
separating any two points of intersection, and so, applying the method ad
infinitum, we should have to conclude ultimately that the circle with radius
CH (or CG) coincides with AB, It would follow that a drcle with centre
C and radius greater than CH would not meet AB at all. Also, since all
straight lines from C to points on AB would be equal in length, there would
be an infinite number of perpendiculars from C on AB.

Is this under any circumstances possible? It is not possible in Euclidean
space, but it is possible, under the Riemann hypothesis (where a straight line
is a "closed series" and returns on itself), in the case where C is the pole of
the straight line AB.

It is natural therefore that, for a proof that in Euclidean space there is
only one perpendicular firom a point to a straight line, we have to wait until
1. 16, the precise proposition which under the Riemann hypothesis is only valid
with a certain restriction and not universally. There is no difficulty involved
by waiting until i. 16, since i. 12 is not used before that proposition is reached ;
and we are only in the same position as when, in order to satisfy ourselves of
the number of possible solutions of i. i, we have to wait till i. 7.
I But if we wish, after all, to prove the truth of the assumption without
recourse to any later proposition than i. 12, we can do so by means of this
same invaluable i. 7.

H. E. 18

«74

BOOK I

If the circle intersects AB as before in tr, jE^ let i7be the middle point of
GE^ and suppose, if possible, that the
circle also intersects AB in any other point
AT on ^^.

From H^ on the side of AB opposite to
C, draw HL at right angles to AB^ and
make HL equal to HC.

Join CG, LG, CK, LK.

Now, in the tnangles CBG^ LHG^
CH'xs equal to LH^ and HG is common.

Also the angles CHG, LHG^ being
both right, are equal.

Therefore the base CG is equal to the base LG.

Similarly we prove that CK is equal to LK.

But, by hypothesis, since A* is on the circle,
CK\& equal to CG.

Therefore CG, CK, LG, LKzxe all equal.

Now the next proposition, i. 13, will tell us that Clf, HL are in a straight
line; but we will not assume this. Join CZ.

Then on the same base CL and on the same side of it we have two pairs
of straight lines drawn from C, L to G and K such that CG b equal to CK
and LG to LK

But this is impossible [i. 7].

Therefore the circle cannot cut BA or BA produced in any point other
than G on that side of CL on which G is.

Similarly it cannot cut AB or AB produced at any point other than E
on the other side of CL>

The only possibility left therefore is that the circle might cut AB in the
same point as that in which CL cuts it But this is shown to be impossible
by an adaptation of the proof of i. 7.

For the assumption is that there may be some point M on CL such that
CM is equal to CG and ZAT to LG.

If possible, let this be the case, and produce CG
toN.

Then, since CM is equal to CG,
the angle NGM is equal to the angle GML [i. 5, part 2].

Therefore the angle GML is greater than the angle
MGL.

Again, since LG is equal to LM,
the angle GML is equal to the angle MGL.

But it was also greater : which is impossible.

Hence the circle in the original figure cannot cut AB in the point in
which CL cuts it

Therefore the circle cannot cut AB xn any point whatever except G and E.

[This proof of course does not prove that CK is less than CG, but only
that it is not equal to it The proposition that, of the obliques drawn
from C to AB, that is Jess the foot of which is nearer to ZTcan only be proved
later. The proof by i. 7 also fails, under the Riemann hypothesis, if C, Z are
the poles of the straight line AB, since the broken lines CGL, CKL etc.
become equal straight lines, all perpendicular to AB\

Produs rightly adds (p. 289, 18 sqq.) that it is not neassary to take D on
the side of AB away from A^iwa objector ''says that there is no space on

I. 12, 13] PROPOSITIONS 12, 13 275

that side." If it is not desired to trespass on that side of AB^ we can take D
anywhere on AB and describe the €irc of a circle between D and the point
where it meets AB again, drawing the arc on the side of AB on which C is.
If it should happen that the selected point D is such that the circle only meets
AB in one pomt {D itself), we have only to describe the circle with CD as
radius, then, if ^ be a point on this circle, take /*a point further from C than
E is, and describe with CF as radius the circular arc meeting AB in two
points.

Proposition 13.

If a straight line set up an a straight line make angles^ it
will make either two right angles or angles equal to two right
angles.

\ For let any straight line AB set up on the straight line

s CD make the angles CBA, ABD ;

I say that the angles CBA, ABD
are either two right angles or equal to
two right angles.

Now, if the angle CBA is equal to g

to the angle ABD,

they are two right angles. [Def. 10]

But, if not, let BE be drawn from the point B at right

' angles to CD\ [i. n]

therefore the angles CBE, EBD are two right angles,
fis Then, since the angle CBE is equal to the two angles
CBA, ABE,

let the angle EBD be added to each ;
therefore the angles CBE, EBD are equal to the three
angles CBA, ABE, EBD. , [C. N. 2]

Again, since the angle DBA is equal to the two angles
DBE, EBA,

let the angle ABC be added to each ;
therefore the angles DBA, ABC are equal to the three
angles DBE, EBA, ABC. [C N. 2]

i But the angles CBE, EBD were also proved equal to
the same three angles ;

and things which are equal to the same thing are also
equal to one another ; [C. N. 1]

therefore the angles CBE, EBD are also equal to the
.0 angles DBA, ABC

18—2

276 BOOK I [h 13, 14

But the angles CBB, EBD are two right angles ;

therefore the angles DBA, ABC are also equal to two

right angles.

Therefore etc.

Q. E. D.

17. let the angle EBD be added to each, literally ** let the angle EBD be added
(so as to be) common,'* K9Uf^ wfnontiffBu ^ W6 BBA. Sinularly irouH^ d^g^^ is med of
subtracting a straight line or angle from each of two others. "Let the common angle EBD
be added '^is clearly an inaccurate translation, for the angle is not common before it is added,
i.e. the irouH^ is proleptic. ** Let the common angle be sttkraOad** as a translation of atu^
d^pf^Bw woula be less unsatisfactory, it is true, out, as it is desirable to use correraondhig
words when translating the two expressions, it seems hopeless to attempt to keep toe won
« common,'* and I have therefore said ** to each " and " nrom each " simpfy.

Proposition 14.

If with any straight line, and at a point on it, two stra^kt
lines not lying on the same side make the adjacent angles equal
to two right angles, the two straight lines will be in a straight
line with one another.

5 For with any straight line AB, and at the point B on it,
let the two straight lines BC, BD not lying on the same side
make the adjacent angles ABC, ABD equal to two right
angles ;

I say that BD is in a straight line with CB.
o For, if BD is not in a straight line
with BC, let BE be in a straight line
with CB.

Then, since the straight line AB g —
stands on the straight line CBE,
s the angles ABC, ABE are equal to two right angles.

[1. 13]
But the angles ABC, ABD are also equal to two right angles ;
therefore the angles CBA, ABE are equal to the angles
CBA, ABD. [Post 4 and C. N. i]

Let the angle CBA be subtracted frpm each ;
» therefore the remaining angle ABE is equal to the remaining
angle ABD, [C N. 3]

the less to the greater : which is impossible.
Therefore BE is not in a straight line with CB.
Similarly we can prove that neither is any other straight
\s line except BD.

I. 14, is] propositions 13—15 277

Therefore CB is in a straight line with BD.
Therefore etc. ^ « ^

Q. E. D.

I. If with any straight line.... There is no greater difficulty in translating the works
of the Greek geometers than that of accurately giving the force of prepositions, rp^t, for
Instance, is used in all sorts of expressions with various shades of meaning. The present
enunciation begins *E^ irpbt ripi tidel^ koI rf *-p6t a&ri ffri/^ff% *nd it is rdly necessary in
this one sentence to translate rpdf by three different words, witAt at, and am. The first wp^
must be translated by wit A because two straight lines " makj" an angle vniA one another. On
the other hand, where the similar expression wpits rj BodelffTg tdBtl^ occurs in 1. 33, but it is
a question of "constructing" an angle (o^uon^ao'tfac), we have to sav "to construct ch a
given straight line." Against would perhaps be the English word comine nearest to
expressing all these meanings of «7>6f, but it would be intolerable as a tranuation.

17. Todhunter points out that tor the inference in this line Post. 4, that all right angles
are equal, is necessary as well as the Common Notion that things which are equal to the same
thing (or rather, here, to e^uat tAings) are equal. A similar remark applies to steps in the
pro(»s of 1. 15 and i. 38.

34. we can prove. The Greek expresses this by the future of the verb, de(^>/ier,
** we shall prove,'* which however would perhaps be misleading in English.

Proclus observes (p. 297) that two straight lines on the same side of another
straight line and meeting it in one and the same
point may make with one and the same portion
of the straight line terminated at the pomt two
angles which are together equal to two right angles,
in which case however the two straight lines would
not be in a straight line with one another. And
he quotes from Porphyry a construction for two
such straight lines in the particular case where they
form with the given straight line angles equd
respectively to half a right angle and one and a
half right angles. There is no particular value in
the construction, which will be gathered from the annexed figure where CE^
CF are drawn at the prescribed inclinations to CD.

Proposition 15-

If two straight lines cut one another, they make the vertical
angles equal to one another.

For let the straight lines AB, CD cut one another at the
point E ;

I say that the angle A EC is equal to ^.^^

the angle DEB, ^^>-^

and the angle CEB to the angle o ^^<~c

AED. §

For, since the straight line AE stands
3 on the straight line CD, making the angles CEA, AED,
the angles CEA, AED are equal to two right angles.

[I- 13]

378 BOOK I [l 15

Again, since the straight line DE stands on the straight
line AB, making the angles AED\ DEB,

the angles AED, DEB are equal to two right angles.

IS But the angles CEA, AED were also proved equal to
two right angles ;

therefore the angles CEA, AED are equal to the
angles AED, DEB. * [Post 4 and C A: i]

Let the angle AED'h^ subtracted from each ;
» therefore the remaining angle CEA is equal to the
remaining angle BED. [C. N. 3]

Similarly it can be proved that the angles CEB^ DEA
are also equal.

Therefore etc. q. E. d.

25 [PoRiSM. From this it is manifest that, if two straight
lines cut one another, they will make the angles at the point
of section equal to four right angles.]

I. Uie vertical angles. The diiference between atffacmt angles («l tf^cf^t T wwht) and
vertuai angles {ol irarA Kofio^ >Mr(ai) is thus explained by Produs {p, 198, 14—14). The
first term describes the angles made by two straight lines whfXk one onlv is dividedf by the
other, i.e. when one straight line meets another at a point which is not either of its extremi-
ties, but is not itself produced beyond the point of meeting. When the first straight line is
produced, so that the lines cross at the pomt, they make two pairs of vertuai angles (whidi
are more clearly described as vertically opposite angles), and which are so called because their
converrcnce is from opposite directions to one point (the intersection of the lines) as vertex
(«»pi^?).

36. at the point of section, literally "at the section," rpftt r% ro/ii.

This theorem, according to Eudemus, was first discovered by Thales, but
found its scientific demonstration in Euclid (Proclus, p. 299, 3—^).

Proclus gives a converse theorem which may be stated thus. If a straight
line is met at one and the same point intermediate in its length by two other
straight lines on different sides of it and such as to mahe the vertical angles
equals the latter straight lines are in a straight line with one another. The
proof need not be given, since it is almost self-evident, whether (i) it is dirtet,
by means of i. 13, 14, or (2) indirect, by reductio ad absurdum depending
on I. 15.

The balance of MS. authority seems to be against the genuineness of this'
Porism, but Proclus and Psellus both have it. The word is not here used, as it
is in the title of Euclid's lost Porisms, to si^ify a particular class of independent
propositions which Proclus describes as bemgm some sort intermediate oetween
theorems and problems (requiring us, not to bring a thing into existence, but
to^iM/somethmg which we know to exist). Porism has here (and wherever
the term is used in the Elements) its second meaning ; it is what we call a
corollary, i.e. an incidental result springing from the proof of a theorem or the
solution of a problem, a result not directly sought but appearing as it were by
chance without any additional labour, and constituting;, as Produs sa3rs, a sort
of windfall (^/acuov) and bonus (ic^os). These Ponsms appear in both the

I. 15, 16] PROPOSITIONS IS, 16 279

geometrical and arithmetical Books of the Elements^ and may either result
from theorems or problems. Here the Porism is geometrical, and springs out
of a theorem ; vii. 2 affords an instance of an arithmetical Porism. As an
instance of a Porism to a problem Proclus cites " that which is found in the
second Book'' (ro iv rf Scvr^ fttpkuf ccifuvov) ; but as to this see notes on

II. 4 and IV. 15.

The present Porism, says Produs, formed the basis of ''that paradoxical
theorem which proves that only the following three (regular) polygons can fill
up the whole space surrounding one point, the equilateral triangle, the square,
and the equilateral and equiangular hexagon." We can in fact place round a
point in this manner six equilateral triangles, three regular hexagons, or four
squares. "But only the angles of these regular figures, to the number specified,
can make up four right angles : a theorem due to the Pythagoreans."

Proclus further adds that it results from the Porism that, if any number of
straight lines intersect one another at one point, the sum of all the angles so
formed will still be equal to four right angles. This is of course what is
. generally given in the text-books as Corollaiy 2.

Proposition 16.

In any triangle^ if one of the sides be produced^ the exterior
angle is greaier than either of the interior and opposite angles.
Let ABC be a triangle, and let one side of it BC be
produced to D\
5 I say that the exterior angle ACD is greater than either
of the interior and opposite angles
CBA, BAC.

Let A Che bisected at £ [i. lo],
and let B£ be joined and produced
10 in a straight line to £]

let £Fhe made equal to B£[i. 3],
let FC be joined [Post, i], and let A C
be drawn through to G [Post. 2].
Then, since A£ is equal to £Ct
cs and B£ to £F,

the two sides A£, £B are equal to the two sides C£,
£F respectively ;

and the angle A£B is equal to the angle F£Cy

for they are vertical angles. [i. 15]

JO Therefore the base AB is equal to the base FC,

and the triangle AB£ is equal to the triangle CF£,
and the remaining angles are equal to the remaining angles
, respectively, namely those which the equal sides subtend ; [i. 4J
therefore the angle BAB is equal to the angle ECF.

a8o BOOK I [l i6

; But the angle ECD is greater than the angle ECF\

therefore the angle ACD is greater than the angle BAE.

Similarly also, if BC be bisected, the angle BCG, that is,
the angle ACD [i. 15], can be proved greater than the angle
ABC as well.

Therefore etc. Q. E. d.

1. the exterior angle, literally ** the ontside angle,** ^ ifnh^ ymUa.

2. the interior and opposite angles, rOir irr^ jral dhrtroyrlar 7««Mi)ir.
II. let AC be drawn through to O. The word is &4x^«#, a variatioo on the more

usual iKPifiXili^ew, "let it ht producetL'*
II. CFE, in the text •• FEC'

As is well known, this proposition is not universally tnie under the
Riemann hypothesis of a space endless in extent but not infinite in size. On
this hypothesis a straight line is a *' closed series" and returns on itself; and
two straight lines which have one point of intersection have another point of
intersection also, which bisects the whole lengdi of the straight line measured
from the first point on it to the same point again; thus the axiom of Euclidean
geometry that two straight lines do not enclose a space does not hold. If 4A
denotes the finite length of a straight line measured from any point once
round to the same point again, a A is the distance between the two intersecticHis
of two straight lines which meet Two points A^ B do not determine one
sole straight line unless the distance between them is diffioent firom 2A. ^ In
order that there may only be one perpendicular fix>m a point C to a straight
line AB^ C must not be one of the two " poles " of the straight line.

Now, in order that the proof of the present proposition may be universally
valid, it is necessary that C/* should always fall within the angle ACD so that
the angle ACFmay be less than the angle ACD. But this will not always be
so on the Riemann hypothesis. For, (i) if BE is equal to A, so that BF is
equal to 2 A, FmW be the second point in which BE and BD intersect ; i.e.
/•will lie on CD, and the angle ACF will be egua/ to the angle ACD. In
this case the exterior angle ACD will be e^al to the interior angle BAC> ^
(2) If BE is greater than A and less than 2A, so that BF is greater than 2A 1
and less than 4A, the angle ACF will be greater than the angle ACD, and |
therefore the angle ACD will be less than the interior angle BAC. Thus, e.g.y |
in the particular case of a right-angled triangle, the angles other than the right j
angle may be (i) both acute, (2) one acute and one obtuse, or (3) both obtuse
according as the perpendicular sides are (i) both less than A, (2) one less and
the other greater than A, (3) both greater than A.

Proclus tells us (p. 307, 1 — 12) that some combined this theorem with the
next in one enunciation thus: In any triangle, if one side be produced, the
exterior angle of the triangle is greater than either of the interior and opposite
angles, and any two of the interior angles are less than two right angles, the
combination having been suggested by the similar enunciation of EucUd l 32,
In any trian^, if one of the sides be produced, the exterior angle is equal to the
two interior and opposite angles, and the three interior angles of the trian^ are
equal to two right angUs.

The present proposition enables Proclus to prove what he did not succeed
in estabhshing conclusively in his note on 1. 12/ namely ^BcaX from one point
there cannot be drawn to the same straight line three straight lines equal in UngtK

1. i6, 17]

PROPOSITIONS 16, 17

381

I

let AB, AC, AD be all equal, £, C, D being in a
equal, the angles

For, if possible,
straight line.

Then, since AB, AC are

ABC, ACB are equal.
Similarly, since AB, AD are equal, the angles

ABD, ADB are equal.
Therefore the angle ACB is equal to the angle

A DC, i.& the exterior angle to the interior and
opposite angle: which is impossible.

Proclus next (p. 308, 14 sqq.) undertakes to prove by means of i. 16 that,
if a straight line falling on two straight lines make the exterior angle equal to
the interior and opposite angle, the tufO straight lines will not form a triangle or
meet, for in that case the same angle would be both greater and equal.

The proof is really equivalent to that of EucL i. 27. V BE falls on the
two straight lines AB, CD in such a wa^ that the angle
CDE is equal to the interior and opposite angle ABD,
AB and CD cannot form a triangle or meet. For, if
they did, then (by i. 16) the angle CDE would be
greater than the an^le ABD, while by the hypothesis
it is at the same time equal to it

Hence, says Proclus, in order that BA^ DC may
form a triangle it is necessary for them to approach one
another in the sense of being turned round one pair of
corresponding extremities, e.g. B, D, so that the other extremities A, C come
nearer. This may be brought about in one of three ways: (i) AB may
remain fixed and CD be turned about D so that the angle CDE increases ;
(2) CD may remain fixed and AB be turned about B so that the angle ABD
becomes smaller; (3) both AB and CD may move so as to make die angle
ABD smaller and the angle CDE larger at the same time. The reason, then,
of the straight lines AB, CD coming to form a triangle or to meet is (says
Proclus) the movement of the straight lines.

Though he does not mention it here, Proclus does in another passage
(p. 371, 2 — 10, quoted on p. 207 above) hint at the possibiUty that, while i. 16
may remain universally true, either of the straight ?ixit& BA, DC (or both
together) may be turned through any angle not greater than a certain finite
angle and yet may not meet (the Bolyai-Lobachewsky hypothesis).

r

Proposition 17.

In any triangle two angles taken together in any manner
-are less than two right angles.
Let ABC be a triangle ;

I say that two angles of the triangle ABC taken together in

any manner are less than two right angles.

For let BC be produced to I). [Post. 2]

Then, since the angle A CD is an exterior angle of the

triangle ABC,

it is greater than the interior and opposite angle ABC

[1. 16]

282 BOOK I [l 17

Let the angle ACB be added to each ;
therefore the angles A CD, ACB are greater than the angles
ABC, BCA.

But the angles A CD, ^C^^raj equal to two right angles.

[I. «3]

Therefore the angles ^/^|^ BCA are less than two right
imilarly we can prove thitoythe angles BAC, ACB are

angles. f

we can prove uiiMyuic au^ica ajxt,\^, xt,\^aj mc

th an Ji^o right^anglc^and so are the angles CAB,

AiclSm^.,^ pi %.

Thereiore>6tc. ">'sk*.,.^

I ^^^^""^^^^ ^* ^' ^'

I. taken Sogether in any manner, vd^rT^MraXaM/fftv^^MFai, Le. any pair added
togetlier. V

As in his note on the previous propodtiony Produs tries to state the comu
of the property. He takes the case of two straight lines forming right anises
with a transversal and observes that it is the convergence of the straight lifus
towards one another (irvvcvo'if rwf cv^coSy), the lessening oi the two right angles,
which produces the triangle. He will not have it that the &ct of the exterior
angle being greater than the interior and opposite angle is the cause of the
property, for the odd reason that '' it is not necessary that a side should be
produced, or that there should be any exterior angle constructed... and how can
what is not necessary be the cause of what is necessary?" (p. 311, 17 — 21).

Agreeably to this view, Proclus then sets himself to prove the theorem
without producing a side of the triangle.

Let ABC be a triangle. Take any point D on
BC, and join AD.

Then the exterior angle ADC of the triangle ABD
is greater than the interior and opposite angle ABD.

Similarly the exterior angle ADB of the triangle
ADC is greater than'tlie interior and opposite angle
ACD.

Therefore, by addition, the angles ADB, ADC are together greater than
the angles ABC, ACB.

But the angles ADB, ADC are equal to two right angles ; therefore the
angles ABC, ACB are less than two right angles. -

Lastly, Proclus proves (what is obvious from this proposition) that there
cannot be more than one perpendicular to a straight line from a point without
it. For, if this were possible, two of such perpendiculars would form a triangle
in which two angles would be right angles: which is impossible, since any two
angles of a triangle are together less tbuui two right angles.

I. i8]

PROPOSITIONS 17, 18

Proposition 18.

283

In atiy triangle the greater side subtends the greater angle.

For let ABC be a triangle having the side AC greater
than AB ;

I say that the angle ABC is also greater than the angle
BCA.

For, since AC\^ greater than AB, let AD be made equal
to AB [i. 3], and let BD be joined.

Then, since the angle ADB
is an exterior angle of the triangle
BCD,

it is greater than the interior
and opposite angle DCB. [l 16]

But the angle ADB is equal
to the angle ABD,

since the side AB is equal to AD ;

therefore the angle ABD is also greater than the angle
ACB]

therefore the angle ABC is much greater than the angle
ACB.

Therefore etc.

Q. E. D.

In the enunciation of this proposition we have ^orc/yecv (''subtend 'O used with the
simple accusative instead of the more usual inrh with accusative. The latter construction
is used in the enunciation of i. 19, which otherwise only differs from that of 1. 18 in the order
of the words. The point to remember in order to distinguish the two is that the daium
comes first and the quauitum second, the datum being in this proposition the greater side
and in the next the greater angie. Thus the enunciations are (l. 18) wwnht rpiyiiifov if /i/d{taif
wXtvpik rifif fktl^wa yw^lnM intvrthti and (l. 19) waanht rpvyfiifw ^6 rV jul^a ytfifUuf ^
tMtl{m irXcvpd ^or€h€t. In order to keep the proper order in En^lidi we must use the
passive of the verb in i. 19. Aristotle quotes the result of i. 19, using the exact wording,
ord Tdp "Hiw fuli^ ytfiflaw inrorc^ci (Afeteorologica III. 5, 376 a is).

''In order to assist the student in remembering which of these two
propositions [i. 18, 19] is demonstrated directly and
'" which indirectly, it may be observed that the order is
similar to that in i. 5 and i. 6" (Todhunter).

An alternative proof of i. 18 given by Porphjny
(see Proclus, pp. 315, 11 — 316, 13) is interesting. It
starts by supposing a length equal to A£ cut off from
the other end of -^C; that is, CD and not AD is
made equal to AB.

Produce AB to E so that BE is equal to AD, and
join EC

Then, since AB is equal to CD, and BE to AD,
AE is equal to AC

384 BOOK I [l 18, 19

Therefore the angle ARC is equal to the angle ACE.

Now the angle ABC is greater than the angte AEC^ [l 16]

and therefore greater than the angle ACE.
Hence, a fortiori^ the angle ABC is greater than the angle ACB.

Proposition 19.

In any triangle the greater angle is subtended by the
greater side.

Let ABC be a triangle having the angle ABC greater
than the angle BCA ;

I say that the side AC is also greater than the side AB.

For, if not, -^C is either eaual to AB or less.

Now -^C is not equal to AB ;
for then the anc^le ABC would also have been
equal to the angle ACB ; [i. sj

but it is not ;

therefore AC is not equal to AB.

Neither is -^C less than AB,
for then the angle ABC would also have been less than the
Single ACB; [1. iS]

but it is not ;

therefore AC is not less than AB.

And it was proved that it is not equal either.
Therefore -^C is greater than AB.

Therefore etc. Q. e. d.

This proposition, like i. 6, can be proved by merely /dguai deduction from
I. 5 and I. 18 taken together, as pointed out by De Morgan. The seneral
form of the argument u^ by De Morgan is given in his Format Zcgic {iS4j\
p. 25, thus :

^^ Hypothesis. Let there be any number of propositions or assertions —
three for instance, X, Y and Z— of which it is tiie property that one or the
other must be true, and one only. Let there be three other propositions
Py Q and ^ of which it is also the property that one, and one only, must be
true. Let it be a connexion of those assertions that :

when X is true, P is true,

when yis true, Q is true,

when Z is true, ^ is true.
Consequence: then it follows that,

when P)S true, X is true,

•when Q is true, K is true,

when R is true, Zis true."

I

I

[1

5]

[I-

i8]

['

.6]

[I-

^9]

I. 19] PROPOSITIONS 18, 19 385

To apply this to the case before us, let us denote the sides of the triangle
ABC by tf, by r, and the angles opposite to these sides by A^ B^ C respectively,
and suppose that a is the base.

Then we have the three propositions,

when b is equal to r, ^ is equal to C,
when b is greater than r, B is greater than C, 1
when b is less than r, B is less than C, /

and it follows lagUally that,

when B is equal to C, b is equal to r,
\ when B is greater than C, ^ is greater than r, 1

when B is less than C, ^ is less than e. /

Reductio ad absurdum by exhaustion.-

Here, says Proclus (p. 318, 16 — 23), Euclid proves the impossibili^ "by
means of division^* {U huupkr^ioi). This means simply the separation of
different hypotheses, each of which is inconsistent with the truth of the
theorem to be provcKl, and which therefore must be successively shown to be
impossible. If a straight line is not greater than a straight line, it must be
either equal to it or less ; thus in a reductio ad absurdum intended to prove
such a theorem as i. 19 it is necessary to dispose successively of Hue hypotheses
inconsistent with the truth of the theorem.

Alternative (direct) proof.

Proclus gives a direct proof (pp. 319 — 321) which an-Nairia also has and
attributes to Heron. It requires a lemma and is consequently open to the
slight objection of separating a theorem from its converse. But the lemma
and proof are worth giving.

Lemma.

Jfan angle of a triangle be bisected and the straight line bisecting it meet the
base and dMde it into unequal parts^ the sides containing the angle will be
uncqucUy and the greater will be that which meets the greater segment of the base^
and the las thcU which meets the less.

Let ADy the bisector of the angle A of the triangle ABC^ meet BC in Z>,
making CD greater than BD,

I say that AC is greater than AB.
Produce AD ioE so that DE is equal to
AD. And, since Z>C is greater than BD^ cut
oflDFeqvaltoBD.

Join EFand produce it to G.
Then, since the two sides AD, DB are
equal to the two sides ED^ DF^ and the
vertical angles at D are equal,

AB is equal to EF^
and the angle DEF io the angle BAD,

i.e. to the angle DAG (by hypothesis).
Therefore AG\s equal to EG^ [1. 6]

and therefore greater than EF, or AB.
Hence, afartiori, AC\& greater than AB.

U

286 BOOK I [i. 19, to

Proof of I. zg.

Let ABC be a triangle in which the angle ABC is greater than the ang^
ACB*

Bisect BC at Z>, join AD^ and produce it to j? so that DE is equal to
AD, ]o\ti BE.

Then the two sides BD^ DE are equal to the two
sides CD^ DA^ and the vertical angles at Z> are equal ;

therefore BE is equal to AC^

and the angle DBE to the angle at C

But the angle at C is less than the angle ABC\

therefore the angle DBE is less than the angle
ABD.

Hence, if BF bisect the angle ABE^ BF meets
AE between A and D. Therefore EF is greater
than FA.

It follows, by the lemma, that BE is greater than
BA,

that is, AC is greater than AB.

Proposition 20.

In any triangle two sides taken together in any manner
are greater than the remaining one.

For let ABC be a triangle ;
I say that in the triangle ABC two sides taken together in
any manner are greater than the remaining one, namely
BAy AC greater than BC,
AB, BC greater than AC,
BCy CA greater than AB.
For let BA be drawn through to the point /?,
let DA be made equal to CA, and let DC be
joined.

Then, since DA is equal to AC,

the angle ADC is also equal to the angle

ACD; [I.S]

therefore the angle BCD is greater than

the angle ADC. [C N. 5]

And, since DCB is a triangle having the angle BCD
greater than the angle BDC,

and the greater angle is subtended by the greater side,

[i. 19]
therefore DB is greater than BC.

I. 20] PROPOSITIONS 19, ao 287

But DA is equal to AC;

therefore BA, AC are greater than BC.
Similarly we can prove that AB, BC are also greater
than CA, and BC, CA than AB.
Therefore etc.

Q. E, D.

It was the habit of the Epicureans, says Proclus (p. 323), to ridicule this
theorem as being evident even to an ass and requiring no proof, and their
all^ation that the theorem was ''known" (yvwpifiov) even to an ass was based
on the fact that, if fodder is placed at one angular point and the ass at another,
he does not, in order to get to his food, traverse the two sides of the triangle
but only the one side separating them (an argument which makes Savile exclaim
that its authors were "digni ipsi, qui cum Asino foenum essent," p. 78).
Proclus replies truly that a mere perception of the truth of the theorem is a
different thing from a scientific proof of it and a knowledge of the reason why
it is true. Moreover, as Simson says, the number of axioms should not be
increased without necessity.

Alternative Proofs.

Heron and Porphyry, we are told (Proclus, pp. 323 — 6), proved this
tfieorem in different ways as follows, without producing one of the sides.

J^rs/ proof.

Let ABC be the triangle, and let it be required to prove that the sides
BA, AC are greater than BC

Bisect the angle BA C by AD meeting BC in D.

Then, in the triangle ABD,
. the exterior angle ADC is greater than the
interior and opposite angle BAD, [i. 16]

that is, greater than the angle DAC

Therefore the side ^C is greater than the side
CD. [i. 19]

Similarly we can prove that AB is greater than BD,

Hence, by addition, BA, ACsoregjreaittr than BC.

Second proof .

This, like the first proof, is direct. There are several cases to be considered.

ii) If the triangle is equilateral, the truth of the proposition is obvious.
2) If the triangle is isosceles, the proposition needs no proof in the case
(a) where each of the equal sides is ^eater than the base.

(b) If the base is greater than either of the other sides, we have to prove
that the sum of the two equal sides is greater than
the base. Let BC be the base in such a triangle.

Cut off from BC a length BD equal to AB, and
join AD.

Then, in the triangle ADB, the exterior angle
ADC is greater than the interior and opposite angle
BAD. • [i. 16I

Similarly, in the triangle ADC, the exterior angle ADB is greater than the
interior and opposite angle CAD.

388 BOOK I [l 3o

By addition, the two angles BDA^ ADC are together greater than the
two angles BAD, DAC (or the whole angle BAC).

Subtracting the equal angles BDA, BAD^ we have tfie angle ADC
greater than the angle CAD.

It follows that ^C is greater than CD\ [i. 19]

and, adding the equals AB^ BD respectively, we have BA^ AC together
greater than BC

(3) If the triangle be scakm^ we can arrange the sides in order of lensth.
Suppose BC is the greatest, AB the intermediate and AC the least side.
Then it is obvious that AB, BC are together greater than AC^ and BC^ CA
together greater than AB,

It only remains therefore to prove that CA, AB are together greater
than BC.

We cut off from BC a length BD equal to the adjacent side^ join AD^ and
proceed exactly as in the above case of the isosceles trian^e.

Thirdpraof.

This proof is by rtductio ad absurdum.

Suppose that BC is the greatest side and, as before, we have to prove that
BA, ACsLre greater than BC.

If they are not, they must be either equal to
or less than BC

(i) Suppose BA, AC are together equal
to BC.

From BC cut off BD equal to BA, and
join AD.

It follows from the hypothesis that DC is equal to AC.

Then, since BA is equal to BD,
the angle BDA is equal to the angle BAD.

Similarly, since ^C is equal to CD,
the angle CDA is equal to the angle CAD.

By addition, the angles BDA, ADC are together equal to the whole angle
BAC.

That is, the angle BAC is equal to two right angles : which is impossible.

(2) Suppose BA, AC are together less than BC.

From BC cut off BD equal to BA, and from CB cut off CE equal to
CA. Join AD, AB.

In this case, we prove in the same way that
the angle BDA is equal to the angle BAD, and
the angle CEA to the angle CAE.

By addition, the sum of the angles BDA,
AEC is equal to the sum of the angles BAD,
CAE.

Now, by I. 16, the angle BDA is greater than the angle DAC, and
theiefore, a fariiari, greater than the angle EAC.

Similarly the angle AEC is greater than the angle BAD. S

Hence the sum of the angles BDA, AEC is greater than the sum of the '
angles BAD, EAC.

But the former sum was also equal to the lAtter : which is impossible.

! I. ai] PROPOSITIONS 20, 21 389

Proposition 21.

If on one of the sides of a triangle, from its extremities,

there be constructed two straight lines meeting within the

triangle, the straight lines so constructed will be less than the

remaining two sides of the triangle, but will contain a grecUer

I angle.

j On BC, one of the sides of the triangle ABC, from its

extremities B, C, let the two straight lines BD, DC be con-
I structed meeting within the triangle ;

I I say that BD, DC are less than the remaining^ two sides

MO of the triangle BA, AC, but contain an angle BvC greater
I than the angle BAC

For let BD be drawn through to E.

Then, since in any triangle two

, sides are greater than the remaining

ii5 one, [i. 20]

therefore, in the triangle ABE, the

\ two sides -^jff, AE are greater than BE.

Let EC be added to each ;

therefore BA, AC are greater than BE, EC
|ao Again, since, in the triangle CED,
\ the two sides CE^ ED are greater than CD,

\ let DB be added to each ;

therefore CE, EB are greater than CD, DB.
But BA^ AC were proved greater than BE, EC;

therefore BA, AC are much greater than BD, DC.
Agrain, since in any triangle the exterior angle is greater
than the interior and opposite angle, [i. 16]

therefore, in the triangle CDE,

the exterior angle BDC is greater than the angle CED.
o For the same reason, moreover, in the triangle ABE also,
the exterior angle CEB is greater than the angle BAC.
But the angle BDCvias proved greater than the angle CEB;
therefore the angle BDC is much greater than the angle
BAC.
»^ Therefore etc. q. e. d.

3. be constructed... meeting within the triangle. The word "meeting" is not in
the Greek, where the words are irrhs cveraBCMtv, ffwlffroffSai is the word lued of con-
structing two straight lines to a point (cf. i. 7) or^ as to form a triangle ; but it is necessaiy
in Engksh to indicate that they nuet.

3. the straight lines so constructed. Observe the elegant brevity of the Greek ol

H. B. 19

«90

BOOK I

[i. ti

1

The editors generally call attention to the fiiurt that the lines dnwn within
the triangle in this proposition must be drawn,
as the enunciation says, from the ends of the
side ; otherwise it is not necessary that their
sum should be less than that of the remaining
sides of the triangle. Proclus (p. 337, 1 2 sqq.)
gives a simple illustration.

Let i^^C be a right-angled triangle. Take
any point D on BC^ join DA^ and cut off
from it DE equal to AB, Bisect AE at A
and join FC.

Then shall CF, FD be together greater than CA^ AB.
For CF, FE are equal to CF, FA,
and therefore greater than CA.

Add the equals ED, AB respectively ;
therefore CF, FD are together greater than CA, AB.

Pappus gives the same proposition as that just proved, but follows it up
by a number of others more elaborate in character, selected apparently from
" the socalled paradoxes " of one Erycinus (Pappus, in. p. 106 sqq.). Thus
he proves the following :

1. In any triangle, except an equilateral triangle or an isosceles triangle
with base less than one of the other sido, it is possible to construct on the
base and within the triangle two straight lines the sum of which is equal to
the sum of the other two sides of the triangle.

2. In any triangle in which it is possible to construct two straight lines on
the base which are equal to the sum of the other two sides of the triangle it is
also possible to construct two others the sum of which isgnaierXbBn that sum.

3. Under the same conditions, if the base is greater than either of the
other two sides, two straight lines can be constructed in the manner described
which are respectively greater than the other two sides of the triangle ; and the
lines may be constructed so as to be respectively equal to the two sides, if one
of those two sides is less than the other and each of them less than the base.

4. The lines may be so constructed that their sum wiU bear to the sum
of the two sides of the triangle any ratio less than 2:1.

As a specimen of the proofs we will give that of the proposition whidi has

been numbered (i) for the case where the triangle is isosceles (Pappus, iii.
pp. 108 — no).

I. 2i] PROPOSITION 21 391

Let ABC he an isosceles triangle in which the base ^C is greater than
either of the equal sides AB, BC.

With centre A and radius AB describe a circle meeting ACin D.

Draw any radius AEJ^such that it meets ^Cin a point jF outside the circle.

Take any point G on EFy and through it draw (7^ parallel to AC. Take
any point JTon GJfy and draw J^L parallel to IrA meetmg i^C in Z.

From BC cut off -5-A^ equal to EG.

Thus ^^, or LJiiy is equal to the sum of AB^ BN^ and CTV^is less than LK,

Now GFy FHzxe together greater than GH,
and CH, /T^ together greater than CK.

Therefore, by addition,
CA FGy BK9je together greater than CK, HG.

Subtracting HK from each side, we see that
CF, FG are together greater than CK, KG \
therefore, if we add AG to each,

AF, FC are together greater than AG, GK, KC

And AB, BCsLTt together greater than AF, FC. [i. 21]

Therefore AB, BCaie together greater than AG, GK, KC

But, by construction, AB, BN dse together equal to ^^;
therefore, by subtraction, NC is greater than GK, KC,
and a fortiori greater than KC.

Take on KC produced a point Jlfsuch that KM is equal to NC;
with centre A!* and radius JTJl/' describe a circle meeting CL in O, and join KO.

Then shall LK, KO be equal to AB, BC.

For, by construction, ZK is equal to the sum of AB, BN, and KO is
equal to NC\

therefore LK, KO sure together equal to AB, BC

It is after i. 31 that (as remarked by De Morgan) the important
proposition about the perpendicular and obliques drawn from a point to a
straight line of unlimited length is best introduced :

0/ all straight lines that can be drawn to a given straight line of unlimited
length from a given point without it:

(a) the perpendicular is the shortest;

(d) of the obliques, that is the greater the foot of which is further from the
perpendicular ;

{c) given one oblique, only one other can be found of the same length, namely
that the foot of which is equally distant with the foot of the given one from the
perpendicular, but on the other side of it.

Let A be the given point, BC the given straight line; let AD be
the perpendicular from A on BC,
and AE, AF any two obliques of
which AF makes the greater angle
inihAD.

Produce AD to A', making A'D
equal to AD, and join A'E, A F.

Then the triangles ADE, ADE
are equal in all respects ; and so are
the triangles ADF, A'DF

Now (i) in the triangle AEA' the
two sides AE, EA are greater than AA [i. 20J, that is, twice AE is greater
^lan twice AD.

19 — 2

392 BOOK I [mi, 99

Therefore A£ is greater than AD.

(2) Since AE, AE are drawn to £, a point within the triangle AFA\

AF, FA are together greater than AE^ EA\ [l 21]

or twice AFis greater than twice AE.
Therefore ^^is greater than AE.

(3) Along DB measure off DG equal to DF^ and join AG.

The triangles AGD, AFD are then equal in all respects, so that the
angles GAD^ FAD are equal, and AG is equal to AP.

Proposition 22.

Out of three straight lines, which are equal to three given
straight lines, to construct a triangle : thus it is necessary that
two of the straight lines taken together in any manner should
be greater than the remaining one. [i. 20]

Let the three given straight lines be A, B, C, and of these
let two taken together in any manner be greater than the
remaining one,

namely A, B greater than C,

A, C greater than B,
and B, C greater than A ;

thus it is required to construct a triangle out of straight lines
equal to A, B, C.

A-

B-
C-

Let there be set out a straight line DE, terminated at D
but of infinite length in the direction of E,
and let DF be made equal to A, FG equal to B, and GH
equal to C. [i- 3]

With centre F and distance FD let the circle DKL be
described ;

again, with centre G and distance GH let the circle KLH be *
described ;
and let KF, KG be joined ;

I say that the triangle KFG has been constructed out of
three straight lines equal to A, B, C.

i I. at] PROPOSITIONS ai, 22 293

(

(

For, since the point /^ is the centre of the circle DKL,

FD is equal to FK.
But FD is equal to A ;

therefore KF is also equal to A.
Again, since the point G is the centre of the circle LKH^

GH is equal to GK.
But GH is equal to C \

therefore KG is also equal to C
And FG is also equal to B ;
therefore the three straight lines KF, FG, GK are equal to
the three straight lines A, B, C.

Therefore out of the three straight lines KF, FG, GK,
which are equal to the three given straight lines A, B, C, the
triangle KFG has been constructed.

^ Q. E. F.

3 — 4. This is the first case in the Eltments of a iiapifffjM to a problem in the sense of a
statement of the conditions or limits of the possibility of a solution. The criterion is of
coarse supplied by the preceding proposition.

3. thus it is necessary, lliis is usaally translated (e.g. by Williamson and Simson)
**£ui it is necessary,'' which is however inaccurate, since the Greek is not M 84 but 8«t d^.
The words are the same as those used to introduce the tiopw^Lbi in the other sense of the
"definition" or "particular statement** of a construction to be effected. Hence, as in the
latter case we say "thus it is required *' (e.g. to bisect the finite straight line AB, i. lo), we
should here translate *' thus it is necessary.^

4. To this enunciation all the mss. and Boethius add, after the iiOfMiiM^ the words
"because in any triangle two sides taken together in any manner are greater than the
remaining one.** But ub explanation has the appearance of a gloss, and it is omitted by
Produs and Campanus. Moreover there is no corresponding addition to the iwpLrikbt
of VI. «8.

It was early observed that Euclid assumes, without giving any reason, that
the circles drawn as described will meet if the condition that any two of the
straight lines A, B, C are together greater than the third be fulfilled. Proclus
(p- 33I9 S sqq.) argues the matter by means of reductio ad absurdum, but
does not exhaust the possible hypotheses inconsistent with the contention.
He says the circles must do one of three things, (i) cut one another, (2) touch
one another, (3) stand apart (Sicoravat) from one another. He then considers
the hypotheses {a) of their touching externally, (p) of their being separated
from one another by a space. He should have considered also the hypothesis
{f) of one circle touching the other internally or lying entirely within the
other without touching. These three hypotheses being successively disproved,
it follows that the circles must meet (this is the line taken by Camerer and
Todhunter).

Simson says: "Some authors blame Euclid because he does not
demonstrate that the two circles made use of in the construction of this
problem must cut one another : but this is very plain from the determination
he has given, namely^ that any two of the straight lines DF, FG, GHmyi%\.
be greater than the third. For who is so dvdl, though only beginning to
learn the Elements, as not to perceive that the circle described from the
centre F, at the distance FD, must meet FH betwixt F and H, because FD
18 less than FH\ and that, for the like reason, the drcle described from the

394 BOOK I [i. 33, 33

centre G at the distance GH must meet DG betwixt D and G ; and that
these circles must meet one another, because FD and GH are together
greater than FG:'

We' have in fact only to satisfy ourselves that one of the drcles, e.g. that
with centre G^ has at least one point of its circumference outside the other
circle and also at least one point of its circumference inside the same circle ;
and this is best shown with reference to the points in which the first circle
cuts the straight line BE. For ^i) FH^ being equal to the sum of B and C,
is greater than A^ i.e. than the radius of the circle with centre F^ and therefore
H is outside that circle. (2) If GM be measured along (7^ equal to GH
or C, then, since GM is either (a) less or greater than GF, JVwill fidl
either {a) between G and ^or {b) beyond /^towards Z>; in the first case
{a) the sum of FM and C is equal to FG and therefore less than the sum
of A and C, so that FM is less than A or FD; in the second case {i) the
sum of MF and FG^ i.e. the sum of MF and By is equal to GMix C «m1
therefore less than the sum of A^ and B^ so that MF is less than A ^ FD\
hence in either case M &Us within the circle with centre F.

It being now proved that the circumference of the circle with centre G
has at least one point outside, and at least one point inside, the drde with
centre F^ we have only to invoke the Principle of Continm'ty, as we have to
do in I. I (cf. the note on that proposition, p. 242, where the necessary
postulate is stated in the form suggested by Killing).

That the construction of the proposition gives only two points of
intersection between the circles, and therefore only two triangles satisfying
the condition, one on each side of FG^ is dear firom i. 7, which, as bdbre
pointed out, takes the place, in Book i., of iil 10 proving diat two ciides
cannot intersect in more points than two.

Proposition 23.

On a given straight line and at a point an it to construct a
rectilineal angle equal to a given rectilineal angle.

Let AB be the given straight line, A the point on it, and
the angle DCE the given rectilineal angle; #

thus it is required to construct on tne given straight line
AB, and at the point A on it, a rectilineal angle equsu to the
given rectilineal angle DCE.

On the straight lines CD, CE respectively let the points
/?,.£* be taken at random ;
let DE be joined,
arid out of three straight lines which are equal to the three

I. 23] PROPOSITIONS 22, 23 29s

Straight lines CD, DE, CE let the triangle AFG be con-
structed in such a way that CD is equal to AF, CE to AG,
and further DE to FG. ^i. 22]

Then, since the two sides DC, CE are equal to the two

^ sides FA, AG respectively,

' and the base DE is equal to the base FG,

the angle DCE is equal to the angle FAG. [i. 8]

\ Therefore on the given straight line AB, and at the point

I -^ on it, the rectilineal angle FAG has been constructed equal

f to the given rectilineal angle DCE.

This problem was, according to Eudemus (see Proclus, p. 333, 5), "rather
the discovery of Oenopides," from which we must apparently infer, not that
Oenopides was the first to find any solution of it, but that it was he who dis-
covered the particular solution given by Euclid. (Cf. Bretschneider, p. 65.)

The editors do not seem to have noticed the fact that the construction of
the triangle assumed in this proposition is not exactly the construction given
in I. 22. We have here to construct a triangle on a certain finite straight line
AG9& base; in i. 22 we have only to construct a triangle with sides of given
length without any restriction as to how it is to be placed. Thus in i. 22 we
set out any line whatever and measure successively three lengths along it
b^inning from the given extremity, and what we must r^ard as the base is the
intermecUate length, not the length banning at the given extremity of the
straight line arbitrarily set out Here the base is a given straight line abutting
at a given point Thus the construction has to be modified somewhat from

I

ITb

that of the preceding proposition. We must measure AG along AB so that
AG\& equal to CE (or CD), and C^ along GB equal to DE\ and then we
must produce BA, in the opposite direction, to ^ so that AF\& equal to CD
(or CE, ii AG has been made equal to CD).

Then, by drawing circles (i) with centre A and radius AF, (2) with centre
G and radius GIf, we determine J^, one of their points of intersection, and we
prove that the triangle J^AG is equal in all respects to the triangle DCE, and
then that the angle at A is equal to the angle DCE.

I think that Proclus must (though he does not say so) have felt the same
difficulty with regard to the use in i. 23 of the result of i. 22, and that this is
probably the reason why he gives over again the construction which I have
given above, with the remark (p. 334, 6) that "you may obtain the construction
of the triangle in a more instructive manner (StSaoncaXaci^cpov) as follows."

Proclus objects to the procedure of Apollonius in constructing an angle
under the same conditions, and certainly, if he quotes Apollonius correctly, &e
latter's exposition must have been somewhat slipshod.

S96 BOOK I [i. S3» 94

"He takes an angle CDE at random," says Produs (p. 335, 19 sqq.X ''and
a straight line AB^ and with centre D and distance
CD describes the circumference C£, and in die same
way with centre A and distance AB the circumference
FB. Then, cutting off FB equal to CE, he joins AF.
And he declares that the aisles A^ D standii^ on
equal circumferences are equaL" _

In the first place, as Proclus remaiks, it should be E

premised that AB is equal to CD in order that the ^^^

circles may be equal; and the use of Book in. for
such an elementary construction is objectionable.
The omission to state that AB must be taken equal
to CD was no doubt a slip, if it occurred. And, as
r^ards the equal angles '' standing on equal drcum- . ^

.ferences," it would seem possible mat Apollonius said

this in explanation^ for the siJce of brevity, rather than by way of proof. It
seems to me probable that his construction was only given from the point of
view olfrachcal^ not theoretical, geometry. It really comes to the same thing
as Euclid's except that DC is taken equal to DE. For cutting off the arc BF
equal to the arc CE can only be meant in the sense of measuring the M^n/
C£, say, with a pair of compasses, and dien drawing a circle with centre B
and radius eqiial to the chord CE. Apollonius' direction was therefore
probably intended as a practical short cut, avoiding the actual drawing of the
chords CE^ BF^ which, as well as a proof of the ec^uali^ in all respects of the
triangles CDE^ BAF^ would be required to estabhsh tluonikaUy the correct-
ness of the construction.

Proposition 24.

If two triangles have the two sides equal to two sides
respectively^ but have the one of the angles contained by the equal
straight lines greater than the other, they will also have the
bc^e greater than the base.

S Let ABC, DEF be two triangles having the two sides

AB^ -^C equal to the two sides DE, Z?/^ respectively, namely

AB to DE, and -^C to DF, and let the angle at -^ be greater

than the angle at D ;

I say that the base BC is also greater than the base EF.
10 For, since the angle BAC

is greater than the angle EDF,

let there be constructed, on the

straight line DE, and at the

point D on it, the angle EDG
IS equal to the angle BAC; [i. 23]

let DG be made equal to either

of the two straight lines AC,

DF, and let EG, FG be joined.

I

I

' 1.24] PROPOSITIONS 23,24 297

i Then, since AB is equal to DE, and AC to DG,

j ao the two sides BA, AC are equal to the two sides ED, DG,

respectively ;
. and the angle BAC is equal to the angle EDG ;

therefore the base BC is equal to the base EG. [1. 4]
V Again, since DF is equal to DG,

* 25 the angle DGF is also equal to the angle DFG ; [i. 5]

therefore the angle DFG is greater than the angle EGF.
It Therefore the angle EFG is much greater than the angle

* EGF.

And, since EFG is a triangle having the angle EFG
30 greater than the angle EGF,

and the greater angle is subtended by the greater side,

[I. 19]
the side EG is also greater than EF.
But EG is equal to BC.

Therefore BC is also greater than EF.
f 35 Therefore etc.

Q. E. D.

10. I have naturally left out the well-known words added by Simson in
order to avoid the necessity of considering three cases : " Of the two sides
DE, DF let DE be the side which is not greater than the other." I doubt
whether Euclid could have been induced to insert the words himself, even if
it had been represented to him that their omission meant leaving two possible
cases out of consideration. His habit and that of the great Greek geometers
was, not to set out all possible cases, but to give as a rule one case, generally
the most difficult, as here, and to leave the others to the reader to work out for
himself. We have already seen one instance in i. 7.

Proclus of course gives the other
two cases which arise if we do not
first provide that DE is not greater
than DF.

(i) In the first case G may fall
on EF produced, and it is then
obvious that EO is greater than EF,

(2) In the second case EG may
fall below EF.

If so, by 1. 21, DF, FE are
together less than DG, GE.

But DF is equal to DG \ there-
fore EF is less than EG, i.e. than
BC.

These two cases are therefore
decidedly simpler than the case taken
by Euclid as typical, and could well be left to the ingenuity of the learner.

If however after all we prefer to insert Simson's words and avoid the latter

2gB

BOOK I

[1.24

two cases, the proof is not complete unless we show that, with his assumptioo,
/^must, in the figure of the proposition, &11 Mn» EG.

De Morgan would make the following proposition precede: Every siraigki
line drawn from the vertex of a trmngk to the base is less than the greaier of ike
two sides, or than either if they are equal, and he would prove it by means of
the proposition relating to perpendicular and obliques mven above, p. 391.

But it is easy to prove directly that F &lls below EG, if
DE is not greater than DG^ by the method employed by
Pfleiderer, Lardner, and Todhunter.

Let DF, produced if necessary, meet EG in H.

Then the angle DHG is greater than the angle DEG\

[I. 16]

and the angle DEG is not less than the angle DGE :

[.. 18]

therefore the angle DHG is greater than the angle DGH.
Hence DH\% less than DG^ [i. 19]

and therefore DH is less than DF.

Alternative proof.

Lastly, the modem alternative proof is worth giving.

A D A

Let Z>^ bisect die angle FDG (after the triangle DEG has been made
equal in all respects to the triangle ABC^ as in the proposition), and let DH
meet EG in H Join HF.

Then, in the triangles FDH, GDH,

the two sides FD, DHzx^ equal to the two sides GD, DH,
and the included angles FDH, GDHzxt, eqiial ;
therefore the base HF is equal to the base HG.
Accordingly EG is equal to the sum of EH, HF\

and EH, HF are together greater than EF\ [i. 20]

therefore EG, or BC, is greater than EF.
Proclus (p. 339, 1 1 sqq.) answers by anticipation the possible question that
might occur to any one on this propc^ition, viz. why does Euclid not compare
the areas of the triangles as he does in i. 4 ? He observes that inequali^ of
the areas does not follow from the inequality of the angles contained by the
equal sides, and that Euclid leaves out all reference to the question boUi for
this reason and because the areas cannot be compared without the help of the
theory of parallels. *' But if^" says Proclus, *' we must anticipate what is to
come and make our comparison of the areas at once, we assert that (i) if
the an^ Af D — stiptposing that our argument proceeds with reference to the
figure in the proposition — are {together) equeU to two right angles, the triangles

I. 24, 25] PROPOSITIONS 24, 25 299

are proved equals (2) if greater than two right angles^ that triangie which has
the greater angle is lesSy and (3) if they are less, greater" Proclus then gives
the proof, but without any reference to the source from which he quoted
the proposition. Now an-Nairiz! adds a similar proposition to i. 38, but
definitely attributes it to Heron. I shall accordingly give it in the place
where Heron put it

Proposition 25.

// two triangles have the two sides equal to two sides
respectively, but have the base greater than the base, thiy will
also have the one of the angles contained by the equal straight
lines greater than the other.

Let ABC, DEF be two triangles having the two sides
AB^ AC equal to the two sides DE^ Z?/^ respectively, namely
AB to DE, and -^C to DF\ and let the base BC be greater
than the base EF\

I say that the angle BAC is also greater than the angle
EDF.

For, if not, it is either equal to it or less.
• Now the angle BAC is not equal to the angle EDF\

I for then the base BC would also have been equal to the base
, EF, [1.4]

but it is not ;

therefore the angle BAC is not equal to the angle EDF.

Neither again is the angle BAC less than the angle EDF;

for then the base BC would also have been less than the base

EF, ^ [1.24]

but it is not ;
therefore the angle BAC is not less than the angle EDF.
But it was proved that it is not equal either ;

therefore the angle BAC is greater than the angle EDF.
Therefore etc.

Q. E. D.

300 BOOK I [1.95

De Morgan points out that ibis jvoposition (as also i. 8) is a purdy iogieai
consequence of i. 4 and i. 34 in the same way as i. 19 and l 6 are purdy
icgua/ consec^uences of i. 18 and 1. 5. If n, ^, c denote the sidesi A^ B, Cytid ,
angles opposite to them in a triangle ABQ and a^^ V^ /, A\ B, Ciht sides*
and opposite angles respectivdy in a triangle A'BC^ i. 4 and i. 24 tdl us
that, b^ c being respectively equal to V^ i^

(i) if ^ is equal to A\ then a is equal to a\

(2) if ^ is less than A\ then a is less than a',

(3) if ^ is greater than A\ then a is greater than a' ;
and it follows iogually that,

(i) if a is equal to a', the angle A is equal to the angle A\ \u 8]

(3) if a is less than a', ^ is less than A\ \ . .

(3) if a is greater than «*, -^ is greater than A\ ] L'' *5j

Two alternative proofs of this theorem are given by Produs (pp. 345 — 7),
and they are both interesting. Moreover both are Hred.

I. Proof by Menelaus of Alexandria.

Let ABC, jDEFbe two triangles having the two sides BA, AC equal to
the two sides EjD, DF, but the base ^C greater than the base EF.

Then shall the angle at ^ be greater than the angle at Z>.
From BC cut off BG equal to EF. At B, on the straight line BC, make
the angle GBH {on the side of BG remote from A) equal to the angle FED.
Midce BH equal to DE\ join HG, and produce it to meet AC in K.
Join AH.

Then, since the two sides GB, BH are equal to the two sides FE^ ED
respectively,

and the angles contained by them are equal,
HG is equal to DFox AC^

\

and the angle BHG is equal to the angle EDF. ^

Now HK is greater than HG 01 AC, \

and a fortiori greater than AK\

therefore the angle KAH\& greater than the angle KHA.

And« since AB is equal to BH,

the angle BAH\% equal to the angle BHA.

Therefore, by addition,

the whole angle BA C is greater than the whole angle BHG,
that is, greater than the angle EDF.

1. 25, a6] PROPOSITIONS 25, 26 301

II. Heron's proof.

Let the triangles be given as before.

Since BC is greater than EF^ produce EF to (r so that EG is equal to
BC.

Produce ED to If so that DIf is equal to I?F The circle with centre
D and radius Z>^will then pass through If. Let it be described^ as FXIf.

Now, since B:,!^ AC axe together greater than BQ

and BA, AC Bie equal to ED^ />^ respectively,

while BC is eqiial to EG^

EH is greater than EG.

Therefore the circle with centre j^and radius EG will cut Elf^ and

therefore will cut the circle already drawn. Let it cut that circle in JT, and

join DK, KE.

^ Then, since Z> is the centre of the circle FKH^

iDK is eqiial to DF or AC.
Similarly, since E is the centre of the circle KG^
EK\& equal to EG or BC^
And DE is equal to AB.

Therefore the two sides BA^ AC 9st eqiial to the two sides ED^ DK
respectively ;

and the base BC is equal to the base EK\
therefore the angle BAC is equal to the angle EDK.
Therefore the angle BAC is greater than the angle EDF.

Proposition 26.

I If two triangles have the two angles equal to two angles
I respectively y and one side equal to one side^ namely, either the
side adjoining the equal angles, or thai subtending one of the
equal angles, they will also have the remaining sides equal to
5 the remaining sides and the remaining angle to the remaining
angle.

302 BOOK I [l s6

Let ABC, DEF be two triangles having the two angles
ABC, BCA equal to the two angles DEF, EFD respectivelv,
namely the angle ABC to the angle DEF, and the ansle

^^BCA to the angle EFD; and let them also have one side
equal to one side, first that adjoining the equal angles, namely
BC to EF\

I say that they will also have the remaininc; sides equal
to the remaining sides respectively, namely aS to DE and

15-^C to DF, and the remaining angle to the remaining angle,
namely the angle BAC to the angle EDF.

B

For, if AB is unequal to DE, one of them is greater.
Let AB be greater, and let BG be made equal to DE ;
and let GC be joined.
ao Then, since BG is equal to DE, and BC to EF,
the two sides GB, BC are equal to the two sides DE, EF
respectively;
and the angle GBC is equal to the angle DEF;

therefore the base GC is equal to the base DF,
25 and the triangle GBC is equal to the triangle DEF,
and the remaining angles will be equal to the remaining angles,
namely those which the equal sides subtend ; [i. 4]

therefore the angle GCB is equal to the angle DFE.
But the angle DFE is by hypothesis equal to the angle BCA ;
30 therefore the angle BCG is equal to the angle BCA,

the less to the greater : which is impossible.
Therefore AB is not unequal to DE,
and is therefore equal to it.
But BC is also equal to EF-,
35 therefore the two sides AB, BC are equal to the two

sides DE, EF respectively,
and the angle ABC is equal to the angle DEF\

therefore the base ACv& equal to the base /?/%
and the remaining angle BAC is equal to the remaining
40 angle EDF. [l 4]

T

26] PROPOSITION 36 303

Again, let sides subtending equal angles be equal, as AB
, to DE\

' I say again that the remaining sides will be equal to the

remaining sides, namely AC to DF and BC to EF, and

^45 further the remaining angle BAC is equal to the remaining

angle EDF.
1 For, if BC is unequal to EF^ one of them is greater.

Let BC be greater, if possible, and let BH be made equal
to EF\ let Am be joined.
50 Then, since Bm is equal to EF, and AB to DE,
the two sides AB, BH are equal to the two sides DE, EF
respectively, and they contain equal angles ;

therefore the base AH is equal to the base DF,
and the triangle ABH is equal to the triangle DEF,
55 and the remaining angles will be equal to the remaining angles,
namely those which the equal sides subtend ; [i. 4]

therefore the angle BHA is equal to the angle EFD.
But the angle EFD is equal to the angle BCA ;
therefore, in the triangle AHC, the exterior angle BHA is
60 equal to the interior and opposite angle BCA :

which is impossible. [i. 16]

Therefore BC is not unequal to EF,

and is therefore equal to it.
But AB is also equal to DE ;
65 therefore the two sides AB, BC are equal to the two sides
DE, EF respectively, and they contain equal angles ;
therefore the base -^C is equal to the base DF,
the'triangle ABC equal to the triangle DEF,
and the remaining angle BAC equal to the remaining angle
10 EDF. [1.4I

Therefore etc.

rt Q. E. D.

f

1 — 3. the side adjoining the equal angles, vXevpdr rV vp^ Ta7f fo-cuf 'pmUm.

99. is by hjrpothesis equal. Imhtntrnx C^, aocording to the elegant Greek idiom.
iwUutiM is used for the passive of (nrcrrlBiiiu^ as iretMoi is used for the passive of rlBiifu, and
io with the other oompoonds. Cf. vpoffKOffieu, ** to be added.'*

The altemative method of proving this proposition, viz. by applying one
triangle to the other, was very early discovered, at least so far as r^utls the
case where the equal sides are adjacent to the equal angles in each. ^-NairizI
gives it for this case, observing that the proof is one which he had found, but
of which he did not know the author.

ii

304 BOOK I [I. ,6

Proclus has the following interesting note (p. 359, 13 — 18): "Eudemiu
in his geometrical history refers this theorem to Thales. For he says that» in
the method by which they say that Thales proved the distance of ships in the
sea, it was necessary to make use of this theorem." As, wiibrtunatdy, this
information is not sufficient of itself to enable us to determine how Thales
solved this problem, there is considerable room for conjecture as to his
method.

The suggestions of Bretschneider and Cantor agree in the assumption
that the necessary observations were probably made from the top oi some
tower or structure of known height, and that a right-angled triangle was used in
which the tower was the perpendicular, and the line connecting the bottom of
the tower and the ship was the base, as in the annexed figure, where AB is the
tower and C the ship. Bretschneider {Die Geometrie und die Geometer par
EukUides^ % 30) says that it was only necessary for
the observer to observe the angle CAB^ and then
the triangle would be completely determined by
means of this angle and the known length AB.
As Bretschneider says that the result would be
obtained '' in a moment " by this method, it is not
clear in what sense he supposes Thales to have
"observed" the angle BAC. Cantor is more
definite (GescK d. Math, i„ p. 145), for he says that
the problem was nearly related to that of finding the
Seqt from given sides. By the Seqt in the Papyrus Rhind is meant, according
to the conjecture of Cantor and Eisenlohr, a number representing the ratio to
one another of the lengths of certain lines in pyramids or obelisks ; sometimes
it is practically equivalent to the cosine of the angle made by the sloping edge
of a pyramid and the semi-diagonal of the base, sometimes to the tangent of
the angle made by the perpendicular from the vertex of the pyramid on one
side of the base and the line connecting the foot of that perpendicular and the
centre of the base. The calculation of the Seqt thus implying a sort of theory
of similarity, or even of trigonometry, the suggestion of Cantor is apparenUy
that the Seqt in this case would be found from a small right-angled triangle
ADE having a common angle A with ABC as shown in the figure, and that
the ascertained value of the Seqt with the length AB would determine BC.
This amounts to the use of the property of similar triangles; and
Bretschneider's suggestion must apparenUy come to the same thing, since,
even if Thales measured the angle in our sense (e.g. by its ratio to a right
angle), he would, in the absence of something corresponding to a table of
trigonometrical ratios, have gained nothing and would have had to work out
the proportions all the same.

Max C. P. Schmidt also {Kulturhistorische Beitrdge %ur Kenntnis des
griechischen und rihnischen Altertums^ 1906, p. 32) similarly supposes Thales to
have had a right angle made of wood or bronze with the legs graduated, to
have placed it in the position ADE (A being the position of his eye), and
then to have read ofi* the lengths AD, DE respectively, and worked out the
length of BC by rule of three

How then does the supposed use of similar triangles and their property
square with Eudemus' remark about i. 26? As it stands, it asserts the
equality of ttvo triangles which have two angles and one side respectively
equal, and the theorem can only be brought into relation with the above
explanations by taking it as asserting that, if two angles and one side of one
triangle are given, the triangle is completely determined. But, if Thales

I. 26] PROPOSITION 26 305

/ practically Msed froportionsy as supposed, i. 26 is surely not at all the theorem
which this proc^ure would naturally suggest as underlying it and being

( *' necessarily used"; the use of proportions or of similar but not equal

^ triangles would surely have taken attention altogether away from i. 26 and
fixed it on vi. 4.

) For this reason I think Tannery is on the right road when he tries to find

a solution using i. 26 as it stands, and withal as primitive as any recorded
solution of such a problem. His suggestion (La Giomitrie gncque^ pp. 90—1)

* is based on the fluminis varaiio of the Roman agrimensor Marcus Junius
Nipsus and is as follows.

iTo find the distance from a point ^ to an inaccessible point B, From A
measure along a straight line at right angles to AB a
length i^C and bisect it at Z>. From C draw CE at right
angles to CA on the side of it remote from B^ and let E
be the point on it which is in a straight line with B and D,
Then, by 1. 26, CE is obviously equal to AB,
As r^ards the equality of angles, it is to be observed
that those at D are equal beotuse they are vertically
opposite, and, curiously enough, Thales is expressly
^ credited with the discovery of the equality of such angles.
The only objection which I can see to Tannery's
' solution is that it would require, in the case of the ship, a
certain extent of fi-ee and level ground for the construction
and measurements.

I suggest therefore that the following may have been
Thales' method. Assuming that he was on the top of a
tower, he had only to use a rough instrument made of a straight stick and a
cross-piece fastened to it so as to be capable of turning about the fastening
(say a nail) so that it could form any angle with the stick and would remain
where it was put. Then the natural thing would be to fix the stick upright
(by means of a plumb-line) and direct the cross-piece towards the ship.
Next, leaving the cross-piece at the angle so found, the stick could be turned
round, still remaining vertical, until the cross-piece pointed to some visible
object on the shore, when the object could be mentally noted and the distance
^ from the bottom of the tower to it could be subsequently measured. This
! would, by i. 26, give the distance from the bottom of the tower to the ship.
i This solution has the advantage of corresponding better to the simpler and
more probable version of Thales' method of measuring the height of the
pyramids; Diogenes Laertius says namely (i. 27, p. 6, ed. Cobet) on the
authority of Hieronymus of Rhodes (b.c 300 — 260), that he waited for this
purpose until the moment when our shadows are of the same length as ourselves.

Recapitulation of congruence theorems.

Proclus, like other commentators, gives at this point (p. 347, 20 sqq.) a
summary of the cases in which the equality of two triangles m all respects can
be established. We may, he says, seek the conditions of such equality by

(successively considering as hypotheses the eqiiality (i) of sides alone, (2) of
angles alone, (3) of sides and angles combined. Taking (i) first, we can only
establish the equality of the triangles in all respects if all three sides are
respectively equal; we cannot establish the equality of the triangles by any
hypothesis of class (2), not even the hypothesis that all the three angles are
respectively eqiial ; among the hypotheses of class (3), the equality of one

H. E. 20

3o6 BOOK I [i.a6

side and one angle in each triangle is not enough, the equality (a) ci one side
and all three angles is more than enough, as is also the equali^ {i) of two
sides and two or three angles, and {c) of three sides and one or two an^^

The only hypotheses therefore to be eramined firom this point of view aie
the equality of

(a) three sides [EucL l 8].

03) two sides and one angle ^i. 4 proves one case of this, where the an^
is that contained by the sides which are by hypothesis equal].

(y) one side and two angles [i. 36 covers all cases].

It is curious that Proclus makes no allusion to what we call the ambiguams
case^ that case namely of 09) in which it is an angle opposite to one ^tbe
two specified sides in one triangle which is ecjual to the angle opposite to the
equal side in the other triangle. Camerer indeed attributes to Proclus the
observation that in this case the equality of the triangles caimot, unless scmie
other condition is added, be asserted generally; but it would appear that
Camerer was probably misled by a figure (Proclus, p. 351) which looks like a
figure of the ambiguous case but is really only used to show that in l 26 the
equal sides must be camspandrng sides, i.e., they must be either adjacent to the
equal angles in each triangle, or opposite to corresponding equal angles, and
that, e.g., one of the equd sides must not be adjacent to the two angles in
one triangle, while the side equal to it in the other triangle is opposite to one
of the two corresponding angles in that triangle.

The ambiguous case.

If two friangies have two sides equal io two sides respecHveiy^ and if ike
angles opposite to one pair of equal sides he also equals then will ike angles
opposite the other pair of equal sides be either equal or suppUmmtary ; emd^ in
the former case^ the triangles will be equal in all respects.

Let ABC, DEFht two triangles such that AB is equal to DE, and AC
to DE, while the angle ABC is equal to the angle DEF\
it is required to prove that the angles ACB, DFE are either equal or
supplementary.

^ A D

Now (i), if the angle BAC be equal to the angle EDF, it follows, since
the two sides AB^ AC Site equal to the two sides !?£, 27^ respectively, that .
the triangles ABC, DEFdj^ equal in all respects, [i. 4]

and the angles ACB, DFE are equal.

(2) If the angles BAC, EDF be not equal, make the angle EDG (on
the same side of ED as the angle EDF) equal to the angle BAC

Let EF, produced if necessary, meet DG in G.

Then, in the triangles ABC, DEG,
the two angles BAC, ABC are equal to the two angles EDG, DEG
respectively,
and die side AB is equal to the side DE ;

J

I. 26, 27] PROPOSITIONS 26, 27 307

therefore the triangles ABC^ DEG are equal in all respects, [i. 26]
so that the side ^C is equal to the side DG^
and the angle ACB is equal to the angle DGE.
Again, since AC is eqiial to DF9& well as to DG^
DF\& equal to DG,
and therefore the angles DFG^ DGF^xt, equal.

But the angle DFE is supplementary to the angle DFG\ and the angle
DGF^N^s proved equal to the angle ACB;

therefore the angle I?F£ is supplementary to the angle ACB.
If it is desired to avoid the ambiguity and secure that the triangles may
be congruent, we can introduce the necessary conditions into the enunciation,
on the analogy of Eucl. vi. 7.

Jf (tvo triangles have two sides of the one equal to two sides of the other
) respectively^ and the angles opposite to a pair of equal sides equals then, if the
i angles opposite to the other pair of equal sides are both acute^ or hoth obtuse^ or if
one of them is a right angle^ the two triangles are equal in all respects.

(The proof of the three cases (by reductio cui absurdum) was given by
Todhunter.

Proposition 27.

If a straight line falling on two straight lines make the
alternate angles equal to one another, the straight lines mill be
parallel to one another.

% For let the straight line EF falling on the two straight

5 lines AB, CD make the alternate angles AEF, EFD equal
^ to one another ;

I say that AB is parallel to CD.
For, if not, AB, CD when pro-
duced will meet either in the direction
\o of B, D or towardis A, C.

Let them be produced and meet,
in the direction of B, Z>, at G.
* Then, in the triangle GEF,

the exterior angle A EF is equal to the interior and opposite
15 angle EFG :
which is impossible. [i. 16]

Therefore AB, CD when produced will not meet in the
direction of B, D.

Similarly it can be proved that neither will they meet
o towards A, C

20 — 2

i

3o8 BOOK I [i. «7

But straight lines which do not meet in either direction
are parallel ; [Dcf. 23]

therefore AB is parallel to CD.
Therefore etc.

Q. E. D.

I. falling on two straight lines, 4,% Mo MtU% kiarifmntti^ the phnse being the flune
as that used in Post. 5, meaning a trtmsversml*

3. the alternate angles, oi ^miXXAI Twrku. Produt fp. ^57, 9) expfauns that Eadid
uses the word aitemate (or, more exactly, aiiimatefy^ ^roXXd^ m two connexions, (i) of a
certain transformation of a proportion, as in Book V. and the arithmetical Books, (s) as here,
of certain of the angles formed by parallels with a straight line crossing them. AlUnmU
angles are, according to Euclid as interpreted by Prodns, those which are not on the same
side of the transveruil, and are not adjacent, but are separated by the transversal, both being
within the parallels but one ** above ** and the other ** below." The meaning is natoru
enough if we imagine the four internal angles to be taken in cyclic order and aH^nmU angkf
to be any two of them not successive but separated by one angle of the four.'

9. in the direction of B, D or towards A, C, literally ** towards the farts BtD ot
towards A, C," M rii B, A fidpti 41 M rii A, T.

With this proposition b^;ins the second section of the first BodL Up
to this point Eudid has dealt mainly with triangles, their constmcdcm
and their properties in the sense of the relation of their parts, the tides and
angles, to one another, and the comparison of different triangles in respect oi
their parts, and of their area in the particular cases where they are congruent

The second section leads up to the third, in which we pass to rdations
between the areas of triangles, parallelograms and squares, the special feature
being a new conception of equality of areas, equality not dependent on
congruence. This whole subject requires the use of parallels. Consequently
the second section beginning at i. 27 establishes the theory of parallels,
introduces the cognate matter of the equality of the sum of Uie angles of a
triangle to two right angles (i. 32), and ends with two propositions forming the ^
transition to the third section, namely i. 33, 34, which introduce the parallelo- y
gram for the first time.

Aristotle on parallels. ]

We have already seen reason to believe that Euclid's personal contribution '
to the subject was nothing less than the formulation of the famous Postulate
5 (see the notes on that Postulate and on Def. 23), since Aristotle indicates
that the then current theory of parallels contained a petiiio prindpii^ and . !
presumably therefore it was EucUd who saw the defect and proposed the ^
remedy.

But it is clear that the propositions i. 27, 28 were contained in earlier
text-books. They were familiar to Aristotle, as we may judge from two j
interesting passages. I

(i) In Anal. Post i. 5 he is explaining that a scientific demonstration
must not only prove a fact of every individual of a class (#car& wavro^) but '
must prove it primarily and generally true (irp«;^ov #ca^<^Xov) of the whole of
the class as one ; it will not do to prove it first of one part, then of another .
part, and so on, until the class is exhausted. He illustrates this (74 a 13 — 16) I
by a reference to parallels : '' If then one were to show that right (angles) do
not meet, the proof of this might be thought to depend on the fact diat this
is true of all (pairs of actual) right angles. But this is not so, inasmuch as
the result does not follow because (the two angles are) eqtial (to two right

r

1. 27, 28] PROPOSITIONS 27. 28 309

angles^ in the particular way [i.e. because each is a right angle], but by virtue
i of their being equal (to two right angles) in any way whatever [i.e. because
I the sum only needs to be equal to two right angles, and the angles themselves

may vary as much as we please subject to this]."
' (2) The second passage has already been quoted in the note on Def. 23 :

' *' there is nothing surprising in different hypotheses leading to the same false
^ conclusion ; e.g. the conclusion that parallels meet might equally be drawn
I from either of the assumptions (a) that the interior (angle) is greater than the
J exterior or {b) that the sum of the angles of a triangle is greater than two
I right angles" (Anal. Prior, 11. 17, 66 a 11 — 15).

\ I do not quite concur in the interpretation which Heiberg places upon

these passages (Mathematisches tu AristoteleSy pp. 18 — 19). He says, first,
that the allusion to the "interior angle" being "greater than the exterior" in
I the second passage shows that the reference in the first passage must be to
Eucl. I. 28 and not to i. 27, and he therefore takes the words ^i wSl Strot in
the first passage (which I have translated " because the two angles are equal
to two right angles in the particular way ") as meaning " because the angles,
viz. the exterior and the interior^ are equal in the particular way." He also
takes oi ^p^ai ov (rvfixurrovcri (which I have translated " right angles do not
meet," an expression quite in Aristotle's manner) to mean "perpendicular
straight lines do not meet " ; this is very awkward, especially as he is obliged
to supply angles with uroi in the next sentence.

But I think that the first passage certainly refers to i. 28, although I do
not think that the alternative {a) in the second passage suggests it. This
alternative may, I think, equally with the alternative {b) refer to i. 27. That
proposition is proved by rcductio ad absurdum based on the fact that, if the
straight lines do meet, they must form a triangle^ in which case the exterior
angle must be greater than the interior (while according to the hypothesis
these angles are equal). It is true that Aristotle speaks of the hypothesis
that the interior anele is greater than the exterior ; but after all Aristotle had
only to state some incorrect hypothesis. It is of course only in connexion
with straight lines meetings as the hypothesis in 1. 27 makes them, that the
alternative {b) about the sum of the angles of a triangle could come in, and
alternative {a) implies alternative (b).

It seems clear then from AristoUe that i. 27, 28 at least are pre-Euclidean,
^d that it was only in i. 29 that Euclid made a change by using his Postulate.
De Morgan observes that i. 27 is a logical equivalent of i. 16. Thus, if A
means "straight lines forming a triangle with a transversal," B "straight lines
making angles with a transversal on the same side which are together less
than two right angles," we have

All ^ is ^,
and it follows logically that

All not-^ is not-^.

Proposition 28.

If a straight line falling on two straight lines make the
exterior angle equal to the interior and opposite angle on the
same side, or the interior angles on the same side equal to two
right angles, Jhe straight lines will be parallel to one another.

3IO BOOK I [i. 38

For let the straight line EF falling on the two straight
lines AB, CD make the exterior angle EGB equal to the
interior and opposite angle GHD, or the interior angles on
the same side, namely BGH, GHD, equal to two right angles; :

I say that AB is parallel to CD.

For, since the angle EGB is
equal to the angle GHD,
while the angle EGB is equal to the
angle AGH, [i. 15]

the angle AGH is also equal to the
angle GHD\
and they are alternate ;

therefore AB is papdlel to CD. [i. 27]

Again, since the angles BG^.JCrlip^zxt^ equal to two
right angles, and the angles AGH, BuH are also equal to
two right angles, [i. 13]

the angles AGH, BGH are equal to the angles BGH, GHD.

Let the angle BGH be subtracted from each ;
therefore the remaining angle AGH is equal to the remaining
angle GHD\
and they are alternate ;

therefore AB is parallel to CD. [i. 27]

Therefore etc. j

Q. E. D.

One criterion of parallelism, the equality of alternate angles, is given in
I. 37 ; here we have two more, each of which is easily reducible, and is actually
reduced, to the other.

Produs observes (pp. 358 — 9) that Euclid could have stated six criteria as
well as three, by using, m addition, other pairs of angles
in the figure (not adjacent) of which it could be predi-
cated that the two angles are equal or that their sum
is ec^ual to two right angles. A natural division is to
consider, first the pairs which are on the same side of
the transversal, and secondly the pairs which are on
different sides of it

Taking (i) the possible pairs on the satne side, we
may have a pair consisting of

(a) two internal angles, viz. the pairs (BGH,
GHD) and (AGH, GHC) ;

(i) two external angles, viz. the pairs (EGB, DHF) and {EGA, CHF)\

(i) one external and one internal angle, viz. the pairs lEGB^ GHD\
(FHD, HGS), {EGA, GHC) and {FHC, HGA).

I. 28, 29] PROPOSITIONS 28, 29 311

And (2) the possible pairs on different sides of the transversal may consist
respectively of

(fl) two internal angles, viz. the pairs {AGH, GHD) and (CZTG, HGE)\
\b) two external angles, viz. the pairs {AGE, DHF) and {EGB, CHF)\
(f) one external and one internal, viz. the pairs {AGE, GHD\ lEGB,
GHC\ {FHC, HGB) and {FHD, HGA).

The angles are equal in the pairs (i) (r), (2) (a) and (2) {b\ and the sum
is equal to two right angles in the case of the pairs (i) {a\ (i) {p) and (2) (r).
For his criteria Euclid selects the cases (2) (a) [i. 27J and (i) {c\ (i) (a) [i. 28],
leaving out the other three, which are of course equivalent but are not quite
so easily expressed

From Proclus' note on i. 28 (p. 361) we learn that one Aigeias (? Aineias)
of Hierapolis wrote an epitome or abridgment of the Elements, This seems
to be the only mention of this editor and his work; and they are only
mentioned as having combined Eucl. i. 27, 28 into one proposition. To do
this, or to make the three hypotheses the subject of three separate theorems,
would, Proclus thinks, have been more natural than to deal with them, as
Euclid does, in two propositions. Proclus has no suggestion for explaining
Euclid's arrangement unless the ground were that i. 27 deals with angles on
different sides, i. 28 with angles on one and the same side, of the transversal.
But may not the reason have been one of convenience, namely that the
criterion of i. 27 is that actually used to prove parallelism, and is moreover
the basis of the construction of parallels in i. 31, while i. 28 only reduces the
other two hypotheses to that of i. 27, so that precision of reference, as well as
clearness of exposition, is better secured by the arrangement adopted ?

Proposition 29.

A straight line falling on parallel straight lines makes
\ the alternate angles equal to one another, the exterior angle
equal to the interior and opposite angle, and the interior angles
on the same side equal to two right angles.
5 For let the straight line EF fall on the parallel straight
- lines AB, CD ;

I say that it makes the alternate angles AGH, GHD
equal, the ekterior angle EGB equal to the interior and
opposite angle GHD, and the interior angles on the same
10 side, namely BGH, GHD, equal to two right angles.
For, if the angle AGH is unequal
to the angle GHD, one of them is
greater.

Let the angle AGH be greater.
15 Let the angle BGH be added to
each ;

therdfore the angles AGH, BGH
are greater than the angles BGH,
GHD.

1'

31 a BOOK I [h 29

«> But the angles AGIf, BGH are equal to two right angles;

therefore the angles BGH, GHD are less than two
right angles.

But straight lines produced indefinitely from angles less

than two right angles meet ; [Post. 5]

25 therefore AB, CD, if produced indefinitely, will meet ;

but they do not meet, because they are by hypothesis parallel.

Therefore the angle AGH is not unequal to the angle

GHD,

and is therefore equal to it
JO Again, the angle AGH is equal to the angle EGB ; [i. 15]
therefore the angle EGB is also equal to the angle
GHD. [C. N. i]

Let the angle BGH be added to each ;

therefore the angles EGB, BGH are equal to the

js angles BGH. GHD. [C. N. a]

But the angles EGB, BGH are equal to two right angles;

P-I3]
therefore the angles BGH, GHD are also equal to
two right angles.

Therefore etc. q. e. d.

13. straight lines produced indefinitely from angles less than two right angles,
al M dr* IkaffoiufWf if h^ 6p$C» iKfioKKhfUPot, e/f dw€ipow cvftwlwrouavt a variation from the
more explicit language of Postulate 5. A £ood deal is left to be understood, namely that the
straight lines begin from pointo at which they meet a thmsversal, and make with it internal
angles on the same side the sum of which is equal to two right angles.

a6. because they are by hjrpothesis parallel, literally ** because they are supposed
parallel," Ml t6 ra^M^i^Xovf uMls ihrmrfSytfoi.

Proof by " Plasrfair's " axiom.

If, instead of Postulate 5, it is prefen-ed to use '' Playfaiifs " axiom in the
proof of this proposition, we proceed thus.

To prove that the alternate angles AGff, v ^l

GIfD are equal. ^^ ,^- "" p

If they are not equal, draw another straight K- \

line I^L through G making the angle XGIf q Jv^ o

equal to the angle GIfD. \

Then, since the angles KGIf, GIfD are equal, ^

JTZ is parallel to CD. [i. 27]

Therefore ttvo sira^ht lines KL» AB intersecting at G are both parallel to
the straight line CD :

which is impossible (by the axiom).

Therefore the angle ^C?^ cannot but be equal to the angle GHD.

The rest of the proposition follows as in Euclid.

I

1.291 PROPOSITION 29 313

Proof of Euclid's Postulate 5 from " Plasrfair's " axiom.

Let ABy CD make with the transversal EF the angles AEF^ EFC

together less than two right angles.

To prove that AB^ CD meet towards A^ C p _— — -B

Through E draw GH making with EF the angle Q.j^^;:::::;.^^-^^^^^^^— --H

GJS/' equal (and alternate) to the angle EFD, ^ \

Thus G-^is parallel to CD, [i. 27] \

Then (i) AB must meet CD in one direction or ^ F

the other.

For, if it does not, AB must be parallel to CD\ hence we have two

straight lines AB^ GH intersecting at E and both parallel to CD :

which is impossible. ^j2^ v^la-^-^. juy^^ .^v^

Therefore AB^ CD must meet.

(2) Since ABy CD meet, they must form a triangle with EF,

But in any triangle any two angles are together less than two right angles.

Therefore the angles AEF^ EFC (which are less than two right angles),
and not the angles BEF^ EFD (which are together greater than two right
angles, by i. 13), are the angles of the triangle ;

that is, EAy FC meet in the direction of A^ C, or on the side of EF on
which are the angles together less than two right angles.

The usual course in modem text-books which use " Playfair's " axiom in
lieu of Euclid's Postulate is apparently to prove i. 29 by means of the axiom,
and then Euclid's Postulate by means of i. 29.

De Morgan would introduce the proof of Postulate 5 by means of
"Playfair's" axiom before i. 29, and would therefore apparently prove i. 29 as
Euclid does, without any change.

As between Euclid's Postulate 5 and " Playfair's " axiom, it would appear
that the tendency in modem text-books is rather in favour of the latter.
Thus, to take a few noteworthy foreign writers, we find that Rausenberger
stands almost alone in using Euclid's Postulate, while Hilbert, Henrici and
Treutlein, Rouch^ and De Comberousse, Enriques and Amaldi all use
" Playfair's " axiom.

Yet the case for preferring Euclid's Postulate is argued with some force by

Dodgson (Euclid and his modem Rivals^ pp. 44—6). He maintains (i) that

:' Playfair's" axiom in fact involves Euclid's Postulate, but at the same time

I involves tnare than the latter, so that, to that extent, it is a needless strain on

the faith of the learner. This is shown as follows.

Given AB, CD making with EF\}^t, angles AEF, EFC together less than
two right angles, draw GH through E so that the angles GEF, EFC are
together equal to two right angles.

Then, by i. 28, GH, CD are ''separational."

We see then that any lines which have the property (a) that they make
with a transversal angles less than two right angles have also the property {fi)
that one of them intersects a straight line which is '' separational " from
the other.

Now Playfair's axiom asserts that the lines which have property (fi) meet
if produced : for, if they did not, we should have two intersecting straight
lines both '' separational " from a third, which is impossible.

We then argue that lines having property (a) meet because lines having
property (a) are lines having property (fi). But we do not know, until we
have proved i. 29, that all pairs of lines having property (fi) have also property

314 BOOK I [i. 29, 30

(a). For anything we know to the contiary, class (fi) mmy be greater than
class (a). Hence, if you assert anything of dass (/S), the logical ^ect is more
extensive than if you assert it of dass (a) \ for you assert it, not only of that
portion of class (fi) which is known to be included in dass (a), but abo of the
unknown (but possibly existing) portion which is ff^/ so induded. <

(2) Eudid's Postulate pjuU before the beginner clear and posUm con-
ceptions, a pair of straight lines, a transversal, and two angl^ together less
than two right angles, whereas " Playfair's " axiom requires him to reaEse a
pair of straight lines which never meet though produced to infinity : a mgatioe
conception which does not convey to the mmd any clear notion of therdative
position of the lines. And (p. 68) Eudid's Postulate gives a direct criterion
for judging that two straight hnes meet, a criterion which is constantly reqdied,
e^. in I. 44. It is true that the Postulate can be deduced from ^Playftir's''
axiom, but editors frequently omit to deduce it, and then taddy assume it
afterwards : which is the least justifiable course of all

Proposition 30.

Straight lines parallel to the same straight line are also
parallel to one another.

Let each of the straight lines AB^ CD be parallel to EF\
I say that AB is also parallel to CD.
5 For let the straight line GK fall upon
them.

Then, since the straight line GK
has fallen on the parallel straight lines
AB, EF,
o the angle AGK is equal to the

angle GHF. [l 29] j

Again, since the straight line GK has fallen on the parallel I
straight lines EF, CD, -

the angle GHF is equal to the angle GKD. [i. 29]
5 But the angle AGK "w^s also proved equal to the angle
GHF\

therefore the angle AGK is also equal to the angle
GKD ; [C. N. il

and they are alternate.

o Therefore AB is parallel to CD.

Q. E. D.

«o. The usual conclusUn in general terms (" Therefore etc'*) repeatiiig the ennnciatioii
is, curiously enough, wanting at the end of this proposition.

The proposition is, as De Morgan points out, the logical equivalent of
''Playfoir's" axiom. Thus, if X denote "pairs of straight lines intersecting one

30. 3']

PROPOSITIONS 29—31

3>S

another," K^' pairs of straight lines parallel to one and the same straight line,"
we have

No X is K,
and it follows logically that

No Y is X.

De Morgan adds that a proposition is much wanted about parallels ^or
perpendiculars) to two straight lines respectively making the same angles with
one another as the latter do. The proposition may be enunciated thus :

If the sides of one angle be respectively (i) parallel or (2) perpendicular to
the sides of another ang^^ the two angles are either
equal or supplementary.

(i) Let DE be parallel to AB and GEF parallel
to BC.

To prove that the angles ABC^ DEG are equal
and the angles ABC^ JDEF supplementary.

Produce DE to meet BC in H,

Then [i. 29] the angle DEG is equal to the angle
DHC,

and the angle ABC is equal to the angle DHC
Therefore the angle DEG is equal to the angle ABC\ whence also the
angle DEF is supplementary to the angle ABC

(2) Let ED be perpendicular to AB, and GEF perpendicular to BC,

To prove that the angles ABC, DEG are
equal, and the angles ABC, DEF supplementary.

Draw Ejy at right angles to ED on the side
of it opposite to B, and (Uaw EG at right angles
to EF on the side of it opposite to B,

Then, since the angles BDE, DED, being
right angles, are equal,

ED is parallel to BA. [i. 27]
Similarly EG' is parallel to BC

Therefore [Part (i)] the angle DEG is equal to the angle ABC
But, the right angle DED being equal to the right angle GEG, if the
common angle GEuht subtracted,

the angle DEG is equal to the angle DEG ,
Therefore the angle DEG is equal to the angle ABC\ and hence the
angle DEF is supplementary to the angle ABC

Proposition 31.

Through a given point to draw a straight line parallel to a
given straight line.

Let A be the given point, and BC the given straight
line;
thos it is required to draw through the point A a straight

line parallel to the straight line Bi

3i6 BOOK I [i. 31, 3a

Let a point D be taken at random on BC, and let AD be
joined; on the straight line DA,
and at the point A on it, let the ^ —

angle DAE be constructed equal /

to the angle ADC [i. 23] ; and let the

straight
straight

ine AF be produced in a
ine with EA.

I

Then, since the straight line AD falling on the two
straight lines BC, EF has made the alternate angles EAD,
ADC equal to one another,

therefore EAF is parallel to BC. [i. «7]

Therefore through the given point A the straight line
EAF has been drawn parallel to the given straight line BC.

Q. E, F,

Proclus rightly remarks (p. 376, 14 — 20) that, as it is implied in i. 12
that only one perpendicular can be drawn to a straight line from an external
point, so here it is implied that only one straight line can be drawn through a
point parallel to a given straight line. The construction, be it observed,
depends only upon i. 27, and might therefore have come directly after that
proposition, ^^y then did Eucbd postpone it until after i. 29 and L 30?
Presumably because he considered it nec^sary, before giving the constructioii,
to place beyond all doubt the fact that only one such parallel am be drawn.
Proclus infers this fact from i. 30 ; for, he says, if two straight lines could be
drawn through one and the same point parallel to the same straight line, the two
straight lines would \^ parallel, though intersecting at the given point : which
is impossible. I think it is a fair inference that Euclid would have considered
it necessary to justify the assumption that only one parallel can be drawn
by some such argument, and that he deliberately determined that his own
assumption was more appropriate to be made the subject of a Postulate
than tfie assumption of the uniqueness of the parallel

Proposition 32.

In any triangle, if one of the sides be produced, the exterior
angle is equal to the two interior and opposite angles^ and the
three interior angles of the triangle are equal to two right
angles.

Let ABC be a triangle, and let one side of it BC be
produced to D ;

I say that the exterior angle ACD is equal to the two
interior and opposite angles CAB, ABC, and the three
interior angles of the triangle ABC^ BCA, CAB are equal
to two right angles.

I, 32] PROPOSITIONS 31. 32 317

For let CE be drawn through the point C parallel to the
straight line AB. [i. 31]

Then, since AB is parallel to CE,
and AC has fallen upon them,
the alternate angles BAC, ACE are
equal to one another. [i. 29]

Again, since AB is parallel to
CE,

and the straight line BD has fallen upon them,

the exterior angle ECD is equal to the interior and opposite
angle ABC. [i. 29]

But the angle ACE was also proved equal to the angle
BAC\

therefore the whole angle ACD is equal to the two
interior and opposite angles BAC, ABC

Let the angle ACB be added to each ;

therefore the angles ACD, ACB are equal to the three
angles ABC, BCA, CAB.

But the angles ACD, ACB are equal to two right angles;

P- 13]
therefore the angles ABC, BCA, CAB are also equal

to two right angles.

Therefore etc,

Q. E. D.

This theorem was discovered in the very early stages of Greek geometry.
What we know of the history of it is gathered from three allusions found in
Eutocius, Proclus and Diogenes Laertius respectively.

I. Eutocius at the beginning of his commentary on the Conies of
''Apollonius (ed. Heiberg, Vol. 11. p. 170) quotes Geminus as saying that *'the
ancients (oc ^x^^ot) investigated the theorem of the two right angles in each
individual species of triangle, first in the equilateral, again in the isosceles,
and afterwards in the scalene triangle, and later geometers demonstrated the
general theorem to the effect that in any triangle the three interior angles are
equal to two right angles."

2. Now, according to Proclus (p. 379, 2 — 5), Eudemus the Peripatetic
refers the discovery of this theorem to the Pythagoreans and gives what he
affirms to be their demonstration of it This demonstration will be given
below, but it should be remarked that it is general, and therefore that the
"later geometers" spoken of by Geminus were presumably the Pythagoreans,
whence it appears that the ''ancients" contrasted with them must have
belonged to the time of Thales, if they were not his Egyptian instructors.

3. That the truth of the theorem was known to Thales might also
be inferred from the statement of Pamphile (quoted by Diogenes Laertius,
L 24 — 5, p. 6, ed. Cobet) that "he, having learnt geometry from the

3i8 BOOK I [L39

Egyptians, was the first to inscribe a right-angled triang^ in a aide and
sacrificed an ox" (on the strength of it) ', in odier words, he discovered that
the angle in a semicircle is a right angle. No doubt, when this fact wu once
discovered (empinca/fy^ say), the consideration of the two isosceles triangles
having the centre for vertex and the sides of the right angle for bases
respectively, with the help of the theorem of EucL i. 5, also known to
Thales, would easily lead to the conclusion that the sum of the angles of
a right-angled triangle is equal to two right angles, and it could be readfly
inferred that the angles of any triangle were likewise equal to two right angles
(by resolving it into two right-angled triangles). But it is not easy to see how
the property of the angle in a semicircle could h^prwed except (in the revise
order) by means of the equality of the sum of the angles of a right-an^ii
triangle to two right angles ; and hence it is most natural to suppose, with
Cantor, that Thales proved it (if he did prove it) practically as Euclid does
in III. 31, i.e. by means of i. 32 as applied to righi-angled triangles at all events.
If the theorem of i. 32 was proved before Thales' time, or by Thales
himself, by the stages indicated in the note of Geminus, we may be satisfied
that the reconstruction of the argument of the older proof by Hankd
(pp. 96 — 7) and Cantor (i„ pp. 143 — 4) is not far wronj;. First, it must have
be^sn observed that six angles equal to an angle of an eqmlateral triangle would,
if placed adjacent to one another round a common vertex, fill up the whde
space round that vertex. It is true that Produs attributes to the Pythafpreans
the general theorem that only three kinds of regular polygons, the equilateral
trianf^le, the square and the regular hexagon, can fill up the entire space round
a point, but the practical Imowled^ge that equilateral triangles have this
property could hardly have escaped the Egyptians, whether they made floors
with tiles in the form of equilateral triangles or regular hexagons (Allman,
Greek Geometry from Thales to Euciid, p. 13) or joined the ends of adjacent
radii of a figure like the six-spoked wheel, which was their common form of
whed fi'om the time of Ramses II. of the nineteenth Dynasty, say 1300 b.c
(Cantor, i,, p. 109). It would then be dear that six angles equal to an angle
of an equilateral triangle are equal to four right angles, and therefore that the
three angles of an equilateral triangle are equal to two right angles. (It would
be as clear or clearer, from observation of a square divided into two triangles
by a diagonal, that an isosceles right-angled triangle has each of its equal
angles equal to half a right angle, so that an isosceles right-angled triangle
must have the sum of its angles equal to two right angles.) Next, with leffxA
to the equilateral triangle, it could not fail to be observed
that, if AD were drawn from the vertex A perpendicular
to the base BC, each of the two right-angled triangles so
formed would have the sum of its angles equal to two right y

angles ; and this would be confirmed by completing the /

rectanp;le ADCE, when it would be, seen that the rectangle /

(with Its angles equal to four right angles) was divided by /

its diagonal into two equal triangles, each of which had 6
the sum of its angles equal to two right angles. Next it
would be inferred, as the result of drawing the diagonal of any rectangle and
observing the equality of the triangles forming the two halves, that the sum of
the angles of any right-angled triangle is equal to two right angles, and henoe
(the two congruent right-angled triangles being then pla^ so as to form one
isoscdes triangle) that the same is true of any isosceles triangle. Only the
last step remained, namely that of observing that any triangle could be
regurded as the half of a rectangle (drawn as indicated in the next figure), or

L32] PROPOSITION 32 319

simply that any triangle could be divided into two right-angled triangles,

I whence it would be inferred that in general the

I sum of the angles of any triangle is equal to two f --^^^p;— • ;

I right angles. j X \ ^s^^^ j

Such would be the probabilities if we could \ X \. i

absolutely rely upon the statements attributed to \,^__ ! ^J

Pamphile and Geminus respectively. But in fact

there is considerable ground for doubt in both cases.

1. Pamphile's story of the sacrifice of an ox by Thales for joy at his
discovery that the angle in a semicircle is a right angle is too suspiciously like
the similar story told with reference to Pythagoras and his discovery of the
theorem of Eucl. i. 47 (Proclus, p. 426, 6 — 9). And, as if this were not
enough, Diogenes Laertius immediately adds that '* others, among whom is
Apollodorus the calculator (d XoytorcKo^), say it was Pythagoras" (sc. who
*' mscribed the right-angled triangle in a circle "). Now Pamphile lived in the
reign of Nero (a.d. 54 — 68) and therefore some 700 years after the birth of
Thales (about 640 b.c). I do not know on what Max Schmidt bases his
statement {Kulturhistorische Beitrdge%ur Kenntnis des griechischen und romischm
AUertums^ 1906, p. 31) that "other, much older^ sources name Pythagoras as
the discoverer of the said proposition," because nothing more seems to be
known of Apollodorus than what is stated here by Diogenes Laertius. But it
would at least appear that Apollodorus was only one of several authorities
who attributed the proposition to Pythagoras, while Pamphile is alone
mentioned as referring it to Thales. Again, the connexion of Pythagoras with
the investigation of the right-angled triangle makes it a priori more likely
that it would be he who would discover its relation to a semicircle On
the whole, therefore, the attribution to Thales would seem to be more than
doubtful.

2. As regards Geminus' account of the three stages through which the
proof of the theorem of 1. 32 passed, we note, first, that it is certainly not
confirmed by Eudemus, who referred to the Pythagoreans the discovery of the
theorem that the sum of the angles of any triangle is equal to two right
angles and says nothing about any gradual stages by which it was proved.
Secondly, it must be admitted, I think, that in the evolution of the proof as
reconstructed by Hankel the middle stage is rather artificial and unnecessary,

^since, once it is proved that any right-angled triangle has the sum of its angles
ecjual to two ri^ht angles, it is just as easy to pass at once to any scalene
triangle (which is decomposable into two unequal right-angled triangles) as to
the isosceles triangle made up of two congruent right-angled triangles. Thirdly,
as Heiberg has recently pointed out (Mathematisches zu Aristoteles^ p. 20), it
is quite possible that the statement of Geminus from beginning to end is
simply due to a misapprehension of a passage of Aristotle {Anal, Post. i. 5,
74 a 25). Aristotle is illustrating his contention that a property is not
scientifically proved to belong to a class of things unless it is proved to belong
primarily {wfHarov) and generally (iraMXou) to the whole of the class. His first
illustration relates to parallels making with a transversal angles on the same
side together equal to two right angles, and has been quoted above in the note
on I. 27 (pp. 308—9). His second illustration refers to the transformation of
a proportion altemando^ which (he says) ''used at one time to be proved
separately " for numbers, lines, solids, and times, although it admits of being
proved of all at once by one demonstration. The third illustration is : " For
the same reason, even if one should prove (ovS* av rvi Sciif^) with reference to

320 BOOK I [1.32

each (sort of) triangle, the equilateral, scalene and isosceles, separateljr, that
each has its angles equal to two right angles, either by one proof or by different
proofs, he does not yet know that t?u trian^^ i.e. the triangle in gtmrai^ has
Its angles equal to two right angles, except in a sophistical sense, even though
there exists no triangle other than triangles of the kinds mentioned. For he
knows it, not qu& triangle, nor of evtry triansle, except in a numerical sense
(car apiO/ior); he does not know it noHonally (icar cISo«) of every triangle^ even
though there be actually no triangle which he does not know."

The difference between the phrase *' used at one time to be proved " in
the second illustration and '' if any one should prove " in the third appears to I
indicate that, while the former referred to a historical fact, the latter does not;
the reference to a person who should prove the theorem of i. 32 for the three
kinds of triangle separately, and then claim that he had proved it generally, i|
states a purely hypothetical case, a mere illustration. Yet, coming after the j
historical Ibxx stated in the preceding illustration, it might not unnaturally give
the impression, at first sight, that it was historicsil too.

On the whole, therefore, it would seem that we cannot safely go behind
the dictum of Eudemus that the discovery and proof of the theorem of i. 32
in all its generality were Pythagorean. This does not however preclude its
having been discovered by stages such as those above set out after Hankel
and Cantor. Nor need it be doubted that Thales and even his Egyptian
instructors had advanced some way on the same road, so far at all events as
to see that in an equilateral triangle, and in an isosceles right-angled triangle,
the sum of the angles is equal to two right angles.

The Pythagorean proof.

This proof, handed down by Eudemus (Proclus, p. 379, 2 — 15), is no
elegant than that given by EucUd, and is a natural
development from the last figure in the recon-
structed argument of Hankel. It would be seen,
after the theory of parallels was added to geometry,
that the actual drawing of the perpendicular and
the complete rectangle on BC as base was
unnecessary, and that the parallel to BC through
A was all that was required.

Let ABC be a triangle, and through A draw DE parallel to BC [1. 31]

Then, since BC^ DE are parallel,
the alternate angles DAB^ ABCzit equal, [i. 29]

and so are the alternate angles EAd ACB also.

Therefore the angles ABC^ ACB are together equal to the angles
DAB, EAC

Add to each the angle BAC;
therefore the sum of the angles ABC, ACB, BAC is equal to the sum of the
angles DAB, BAC, CAE, that is, to two right angles.

Euclid's proof pre-Euclidean.

The theorem of i. 32 is Aristotle's favourite illustration when he wishes to
refer to some truth generally acknowledged, and so often does it occur that
it is often indicated by two or three words in themselves hardly intelli^ble,
e.g. TO fivcrlv ^p^ois (Ana/, Post, i. 24, 85 b 5) and vwapxti wavrl rpiyiiytf ro fivo
{Md, 85 b 11).

One passage (Metaph. 1051 a 24) makes it clear, as Heiberg (op,*dt.

I. 32] PROPOSITION 32 321

p. 19) acutely observes, that in the proof as Aristotle knew it Euclid's
construction was used. "Why does the triangle make up two right angles?
Because the angles about one point are equal to two right angles. If then the
parallel to the side had been drawn up (avrJKro), the fact would at once have
oeen clear from merely looking at the figure." The words "the angles about
one point" would equally fit the Pythagorean construction, but "drawn
upwards " applied to the parallel to a side can only indicate Euclid's.

Attempts at proof independently of parallels.
The most indefatigable worker on these lines was Legendre, and a sketch
of his work has been given in the note on Postulate 5 above.

One other attempted proof needs to be mentioned here because it has
^ found much favour. I allude to

Thibaut's method.

This appeared in Thibaut's Grundriss der reinen Mathematik, Gottingen
(2 ed. 1809, 3 ed. 181 8), and is to the following effect

Suppose CB produced to D^ and let BD (produced to any necessary extent
either way) revolve in one direction (say
clockwise) first about B into the position
BA^ then about A into the position oi AC
produced both ways, and lastly about C
mto the position CB produced both ways.

The argument then is that the straight
line BD has revolved through the sum of
the three exterior angles of the triangle.
But, since it has at the end of the revolution

assumed a position in the same straight line with its original position, it must
have revolved through four right angles.

Therefore the sum of the three exterior angles is equal to four right
angles;

frx)m which it follows that the sum of the three angles of the triangle is equal
to two right angles.

But it is to be observed that the straight line BD revolves about different
points in it^ so that there is translation combined with rotatory motion, and it
is necessary to assume as an axiom that the two motions are independent, and
^therefore that the translation may be neglected.

Schumacher (letter to Gauss of 3 May, 1831) tried to represent the
rotatory motion graphically in a second figure as mere motion round a point ;
but Gauss (letter of 17 May, 1831) pointed out in reply that he really
assumed, without proving it, a proposition to the effect that " If two straight
lines (i) and (2) which cut one another make angles A\ A" with a straight
line (3) cutting both of them, and if a straight line (4) in the same plane is
I likewise cut by (i^ at an angle A\ then (4) will be cut by (2) at the angle A\
But this proposition not only needs proof, but we may say that it is, in
I essence, the very proposition to be proved" (see Engel and Stackel, Die
Theorie der ParaUelHnien von Euklid bis auf Gauss^ 1895, P* ^3^)*

How easy it is to be deluded in this way is plainly shown by Proclus'
attempt on the same lines. He says (p. 384, 13—21) that the truth of the
theorem is borne in upon us by the help of " common notions " only. " For,
if we conceive a straight line with two perpendiculars drawn to it at its ex-
tremities, and if we then suppose the perpendiculars to (revolve about
their feet and) approach one another, so as to form a triangle, we see that,

H. E. 21

322 BOOK I [i. 32, 33

to the extent to which they converge, they diminish the right an^es wtidb,
they made with the straight line, so that the amomit taken from the rifjtxt
angles is also the amount added to the vertical angle of the- triangle, and the
three angles are necessarily made ec^ual to two right angles." But a moment^* I
reflection shows that, so far from bemg founded on mere " common notiooa,"
the supposed proof assumes, to begin with, that, if the perpendiculars ap-
proach one another ever so litde, they will then form a triangle immediately,
i.e., it assumes Postulate 5 itself; and the fact about the vertical aiigle can only
be seen by means of the equality of the alternate angles exhibited by drawing
a perpendicular from the vertex of the triangle to the base, Le. KparaiMto
eitner of the original perpendiculars.

Extension to polygons.

The two important corollaries added to i. 32 in Simson's edition are given
by Proclus ; but Proclus' proof of the first is different from, and pertiaps
somewhat simpler than, Simson's.

1. The sum of the interior anf^lts of a convex reetilifieai figure is equal to
twice as many right angles as the figure has sides,
less four.

For let one angular point A be joined to all
the other angular points with which it is not con-
nected already.

The figure is then divided into triangles, and
mere inspection shows

(i) that the number of triangles is two less
than the number of sides in the figure,

(2^ that the sum of the angles of all the ^

triangles is equal to the sum of sdl the interior angles of the fi^re. |

Smce then the sum of the angles of each triangle is equal to two right angles,
the sum of the interior angles of the figure is equal to 2 (/r-2) right angles, I
Le. {2n - 4) right angles, where n is the number of sides in the figure. \

2. The exterior angles of any convex rectilineal
figure are together equal to four right cutgles.

For the interior and exterior angles together are
equal to 2n right angles, where n is the numb^ of sides.

And the intenor angles are together equal to
(2ff-4) right angles.

Therefore the exterior angles are together equal to
(bur ri^ht angles.

This last property is already quoted by Aristotle
as true of all rectilineal figures in two passages (Anal,
Ast. I. 24, 85 b 38 and 11. 17, 99 a 19).

Proposition 33.
TAe stra^kt iines joining equal ami parallel straight
lines (at the extremities which are) in the same directions
{respectively) are themselves also equcU and parallel.

Let AB, CD be equal and parallel and let the straight
5 lines AC^ BD join them (at the extremities which are) in the
same directions (respectively) ;

I. 33i 34] PROPOSITIONS 32—34 323

; I say that AC, BD are also equal and parallel.
Let BC be joined.
Then, since AB is parallel to CD,
10 and BC has fallen upon them,

the alternate angles ABC, BCD
are equal to one another. [i. 29]

And, since AB is equal to CD,
and BC is common,
15 the two sides AB, BC are equal to the two sides DC, CB ;
and the angle ABC is equal to the angle BCD ;

therefore the base -^C is equal to the base BD,
and the triangle ABC is equal to the triangle DCB,
and the remaining apgles will be equal to the remaining angles
20 respectively, namely those which the equal sides subtend ; [1. 4]
therefore the angle ACB\% equal to the angle CBD.
And, since the straight line BC falling on the two straight
lines AC, BD has made the alternate angles equal to one
another,
25 ^C is parallel to BD. [i. 27]

And it was also proved equal to it.
Therefore etc. q. e. d.

I. joining... (at the extremities which are) in the same directions (respectively).
I have for clearness' sake inserted the words in brackets though they are not in Uie original
Greek, which has "joining... in the same directions" or "on the same sides,*' heX rd odrd iiipf^
iwi^€vyw6ciuffai. The expression "towards the same parts,'* though usage has sanctioned it,
b perhaps not quite satisfiurtory.

15. DC, CB and 18. DCB. The Greek has **BC, CD*' and " BCD" in these places
respectively. Euclid is not always careful to write in corresponding order the letters denoting
corresponding points in congruent figures. On the contrary, he evidently prefers the alpha-
betical order, and seems to disdain to alter it for the sake of bqginners or others who might
be confused by it. In the case of angles alteration is perhaps unnecessary ; but in the case
^ of trianeles and pairs of corresponding sides I have ventured to alter the order to that which
the mathematician of to-day expects.

This proposition is, as Proclus says (p. 385, 5), the connecting link between
the exposition of the theory of parallels and the investigation of parallelograms.
For, while it only speaks of equal and parallel straight lines connecting those
ends of equal and parallel straight lines which are in the same directions, it
gives, without expressing the fact, the construction or origin of the parallelogram,
so that in the next proposition Euclid is able to speak of " parallel6grammic
areas" without any further explanation.

Proposition 34.

In parallelogrammic areas the opposite sides and angles
are equal to one another, and the diameter bisects the areas.

Let ACDB be a paFallelogrammic area, and BC its
diameter ;

21 — 2

324 BOOK I [i. 34

5 I say that the opposite sides and angles of the parallelogram
ACDB are equal to one another, and the diameter BC
bisects it

For, since AB is parallel to CD,
and the straight line BC has fallen
ioupon them,

the alternate angles ABC, BCD
are equal to one another. [i. 29]

Again, since -<4C is parallel to BD,
and BC has fallen upon them,

15 the alternate angles ACB, CBD are equal to one

another. [l 29]

Therefore ABC, DCB are two triangles having the two
angles ABC, BCA equal to the two angles DCB, CBD
respectively, and one side equal to one side, namely that
20 adjoining the equal angles and common to both of them, BC\
therefore they will also have the remaining sides equal
to the remaining sides respectively, and the remaining angle
to the remaining angle ; [i. 26]

therefore the side AB is equal to CD,
25 and^Cto^A

and further the angle BAC is equal to the angle CDB.
And, since the angle ABC is equal to the angle BCD,
and the angle CBD to the angle ACB,

the whole an^^le ABD is equal to the whole angle A CD.

[C.N.2\

30 And the angle BAC^^s also proved equal to the angle CDB. ^
Therefore in parallelogrammic areas the opposite sides
and angles are equal to one another.

I say, next, that the diameter also bisects the areas.
For, since AB is equal to CD,
35 and BC is common,
the two sides AB, BC are equal to the two sides DC, CB
respectively ;

and the angle ABC is equal to the angle BCD ;
therefore the base AC is also equal to DB,
40 and the triangle ABC is equal to the triangle DCB. [i. 4]
Therefore the diameter BC bisects the parallelogram

ACDB. Q. E. D.

1.34] PROPOSITION 34 325

I. It b to be observed that, when parallelograms have to be mentioned for the first time,
Euclid calls them *'parallelogrammtc areas*' or, more exactly, ''parallelogram*' areas
{wapaXKii\6yp€ifA/ia x^^pfa). The meaning is simply areas bounded bv parallel straight lines
with the further limitation placed upon the term by Euclid that oiAy jeur-Hded figures are so
called, although of course there are certain regular polygons which have opposite sides
parallel, and which therefore might be said to be areas bouiMed by parallel straight lines. We

Sther from Proclus (p. 30^) that the word "parallelogram** was first introduced by Euclid,
It its use was suggestea by i. 33, and that the formation of the word «-a/>aXXiyX(&Y/N^(/Aot
(paralleMined) was analogous to that of M^rfpanitw (straight-lined or rectilineal).

17, 18, 40. DCB and 36. DC, CB. The Greek has in these places ** BCD'* and
••C/>, BC respectively. Cf. note on i. 33, lines 15, 18.

After specif^ng the particular kinds of parallelograms (squares and rhombi)
in which the diagonals bisect the angles which they join, as well as the areas,
and those (rectangles and rhomboids) in which the diagonals do not bisect
the angles, Proclus proceeds (pp. 390 sqq.) to analyse this proposition with
reference to the distinction in Aristotle's AncU. Post (i. 4, 5, 73 a 21 — 74 b 4)
between attributes which are only predicable of every individual thing (icara
iroi^os) in a class and those which are true of \i primarily {tovtov xpc^ov) and
genercUly (ko^oXov). We are apt, says Aristotle, to mistake a proof «aTa
vavros for a proof rovrov xpcorov KoBokxn) because it is either impossible to
find a higher generality to comprehend all the particulars of which the
predicate is true, or to find a name for it (Part of this passage of Aristotle
has been quoted above in the note on i. 32, pp. 319 — 320.)

Now, says Proclus, adapting Aristotle's distinction to theorems^ the present
proposition exhibits the distinction between theorems which are general and
theorems which are not general. According to Proclus, the first part of
the proposition stating that the opposite sides and angles of a parallelogram
are equal is general because the property is only true of parallelograms ; but
the second part which asserts that the diameter bisects the area is not general
because it does not include all the figures of which this property is true, e.g.
circles and ellipses. Indeed, says Proclus, the first attempts upon problems
seem usually to have been of this partial character (/Acpucoircpai), and generality
was only attained by degrees. Thus ''the ancients, after investigating the
fact that the diameter bisects an ellipse, a circle, and a parallelogram
respectively, proceeded to investigate what was common to these cases,"
though " it is difficult to show what is common to an ellipse, a circle and a
parallelogram."

I doubt whether the supposed distinction between the two parts of the
proposition, in point of " generality," can be sustained. Proclus himself admits
that it is presupposed that the subject of the proposition is a quadrilateral^
because there are other figures (e.g. regular polygons of an even number of
sides) besides parallelograms which have their opposite sides and angles
equal; therefore the second part of the theorem is, m this respect^ no more
< general than the other, and, if we are entitled to the tacit limitation of the
1 theorem to quadrilaterals in one part, we are equally entitled to it in the other.

It would almost appear as though Proclus had drawn, the distinction for
the mere purpose of alluding to investigations by Greek geometers on the
general subject of diameters of all sorts of figures; and it may have been these
which brought the subject to the point at which Apollonius could say in the
first definitions at the beginning of his Conies that '* In any bent line^ such as
is in one plane, I give the name diameter to any straight line which, being
drawn from the bent line, bisects all the straight lines (chords) drawn in the
line parallel to any straight line.** The term bent line {KOfLwvktf ypa^'q)
includes, e.g. in Archimedes, not only curves, but any composite line made

3«6

BOOK I

[>• 34, 35

up of straight lines and curves joined together in any manner. It is of course
clear that either diagonal of a parallelogram bisects all lines drawn within the
parallelogram parallel to the omer diagonal

An-Nairizi gives after i. 31 a neat construction for dividing a straight line
into any number of equal parts (ed. Curtze, p. 74, ed. B^thom-Heibeig,
pp. 141 — 3) which requires only one measuranent repeated, together with the
properties of parallel lines including i. 33, 34. As i. 33, 34 are assumed, I
place the problem here. The particular case taken is the problem of dividing
a straight line into three equal parts.

Let AB be the given straight line. Draw AC^ BD at right angles to it
on opposite sides.

An-Nairizi takes AC^ BD of the same
length and then bisects AC at E and BD
at F, But of course it is even simpler to
measure AE^ EC along one perpendicular
equal and of any length, and BF^ FD along
the other also equal and of the same length.

Join ED, CF meeting AB \n G^ H
respectively.

Then shall AG, GH, iET^ all be equal

Draw HK pandlel to AC, or at right
angles to AB.

Since now EC, FD are equal and parallel
ED, CFiJt equal and parallel. [i- 33 J

And HK was drawn parallel to AC.

Therefore ECHK is a parallelogram ; whence KH is equal as well as
parallel to EC, and therefore to EA.

The triangles EAG, KHG have now two angles respectively equal and the
sides AE, /^ equal

Thus the triangles are equal in all respects, and
AG is equal to GH.

Similarly the triangles KHG, FBH are equal in all respects, and
GH is equal to HB.

If now we wish to extend the problem to the case where AB is to be
divided into n parts, we have only to measure (n—\) successive equal lengths
along AC and {n-i) successive lengths, each equal to the others, along BD.
Then join the first point arrived at' on AC to the last point on BD, the
second on ^^C to the last but one on BD, and so on ; and the joining lines
cut AB in points dividing it into n equal piauts.

Proposition 35.

Parallelograms which are on the same base and in the
same parallels are equal to one another.

Let ABCD, EBCF be parallelograms on the same base
BC and in the same parallels AF, BC ;
5 I say that ABCD is equal to the parallelogram EBCF.
For, since ABCD is a parallelogram,

AD is equal to BC. [l 34]

I. 35] PROPOSITIONS 34, 35 3^7

I For the same reason also

I EF is equal to BCy

iio so that AD is also equal to EF\ \C. N, i]

and DE is common ;

! therefore the whole AE is equal to the whole Z?/^

[C. N. 2]

But AB is also equal to DC\ [i. 34]

therefore the two sides EA, AB are equal to the two sides

}SED, DC respectively,

and the angle FDC is equal to
the angle EAB, the exterior to the
interior ; [i. 29]

therefore the base EB is equal
20 to the base FQ

and the triangle EAB will be equal to the triangle FDC.

[1.4]
Let DGE be subtracted from each ;

therefore the trapezium ABGD which remains is equal to
the trapezium EGCF "which remains. [C N. 3]

25 Let the triangle GBC be added to each ;
therefore the whole parallelogram A BCD is equal to the whole
parallelogram EBCF. [C N. 2]

Therefore etc.

Q. E. D.

11. FDC. The text has "/>/-C."

33. Let DOE be subtracted. Euclid speaks of the triangle DGE without any
explanation that, in the case which he takes (where AD^ EF have no point in common),
.^ B£t CD must meet at a point G between the two parallels. He allows tnis to appear from
the figure simply.

Equality in a new sense.

It is important to observe that we are in this proposition introduced for
the first time to a new conception of equality between figures. Hitherto we
have had equality in the sense of congruence only, as applied to straight lines,
angles, and even triangles (cf. i. 4). Now, without any explicit reference to
any change in the meaning of the term, figures are inferred to be equal which
are equal in area or in content but need not be of the same farm. No
definition of equality is anywhere given by Euclid ; we are left to infer its
meaning from the few axioms about " equal things." It will be observed that
in the above proof the " equality " of two parallelograms on the same base
and between the same parallels is inferred by the successive steps (i) of
subtracting one and the same area (the triangle DGE) from two areas equal
in the sense of congruence (the triangles AEB^ ^PC\ and inferring that the
remainders (the trapezia ABGD^ EGCF) are "equal"; (a) of addi^ one and

328 BOOK I [i. 3S

the same area (the triangle GBC) to each of the latter ** equal" trapezia, and
inferring the equality of the respective sums (the two given parallelograms).

As is well known, Simson (after Qairaut) slightly altered the proof in order
to make it applicable to all the three possible cases. The alteration
substituted one step of subtracting congruent areas (the triangles AEB^ DFC)
from one and the same area (the trapezium ABCF) for the iu>o steps above
shown of first subtracting and then adding a certain area.

While, in either case, nothing more is explicitly used than the axioms that,
if equals be added to equals^ the wholes are equal and that, if equals be subtracted
from equals^ the remainders are equals there is the further tacit assumption that
it is indifferent to what part or from what part of the same or equal areas the
same or equal areas are added or subtracted. De Morgan observes that the
postulate "an area taken from an area leaves the same area from whatever
part it may be taken " is particularly important as the key to equality of non-
rectilineal areas which could not be cut into coincidence geometrically.

Legendre introduced the word equivalent to express this wider sense of
equality, restricting the term equal to things equal in the sense of congruent ;
and this distinction has been found convenient.

I do not think it necessary, nor have I the space, to give any account of
the recent developments of the theory of equivalence on new lines represented
by the researches of W. Bolyai, Duhamel, ])e Zolt, Stolz, Schur, Veronese,
Hilbert and others, and must refer the reader to Ugo Amaldi's article Sulla
teoria dell' equivalenza in Questiani riguardanti la geometria eUmentare
(Bologna, 1900), pp. 103 — 142, and to Max Simon, Ober die Entwicklung der
Elcmentar-^ometrie im XIX. fahrhumUrt {Ijtvpiagy 1906), pp. 115 — lao, with
their full references to the literature of the subject I may however rder to
the suggestive distinction of phraseology used by Hilbert (Grundlagen der
Geometrie, pp. 39, 40) :

(i) "Two polygons are called divisibly-equal (zerlegungsgleich) if they can
be divided into z, finite number of triangles which are congruent two and two."

(2) "Two polygons are called equal in content {inhaltsgleich) or of equal
content if it is possible to add divisibly-equal polygons to them in such a way
that the two combined polygons are ditnsibly-equaiy

(Amaldi suggests as alternatives for the terms in (i) and (2) the expressions
equivalent by sum and equivalent by difference respectively.)

From these definitions it follows that " by combining divisibly-equal '
polygons we again arrive at divisibly-equal polygons; and, if we subtract
divisibly-equal polygons from divisibly-equal polygons, the polygons remaining
are equal in content"

The proposition also follows without difficulty that, "if two polygons are
divisibly-equal to a third polygon, they are also divisibly-equal to one another ;
and, if two polygons are equal in content to a third polygon, they are equal in
content to one another."

The difTerent cases.

As usual, Proclus (pp. 399 — 400), observing that Euclid has given only the
most difficult of the three possible cases, adds the other two with separate
proofs. In the case where E ii) the figure of the proposition falls between A
and A he adds the congruent triangles ABE^ DCF respectively to the
smaller trapezium EBCD^ instead of subtracting them (as Simson does) from
the larger trapezium ABCF,

1.35] PROPOSITION 35 339

An ancient ** Budget of Paradoxes."

Proclus observes (p. 396, 12 sqq.) that the present theorem and the
similar one relating to triangles are among the so^^alled iMiradoxical theorems
of mathematics, since the uninstructed might well regard it as impossible that
the area of the parallelograms should remain the same while the length of the
sides other than the base and the side opposite to it may increase indefinitely.
He adds that mathematicians had made a collection of such paradoxes, the
so-called treasury of paradoxes (6 irofxiSo^os rimoi) — cf. the similar expressions
fifWQii dvaXv6fi€voi (treasury of analysis) and rotros d/(rrpovofMVfitvo9 — in the same
way as the Stoics with their illustrations {Zavtp ol awo rSj$ Sroas hrl rmv
&cy/iar«t>v). It may be that this treasury of paradoxes was the work of
Erycinus quoted by Pappus (iii. p. 107, 8) and mentioned above (note on
I. 31, p. 290).

Locus-theorems and loci in Greek geometry.

The proposition i. 35 is, says Proclus (pp. 394 — 6), the first locus-theorem
(rorucov ^coipi^/ia) given by Euclid. Accordingly it is in his note on this
proposition that Proclus gives us his view of the nature of a locus-theorem
and of the meaning of the word locus (roVos) ; and great importance attaches
to his words because he is one of the three writers (Pappus and Eutocius
being the two others) upon whom we have to rely for all that is known of the
Gredc conception of geometrical loci.

Proclus' explanation (pp. 304, 15 — 395, 2) is as follows. "I call those
(theorems) locus-theorems (roimra) in which the same property is found to exist
on the whole of some locus (irpoc oh^ rm, r6iw^\ and (I call) a locus a position
of a line or a surface producing one and the same property (ypofifi^ 1j iwi-
^vtCoLS $€<nv voiovcrav Iv icai ravrov crv/utima/ia). *For, of locus-theorems, some
are constructed on lines and others on siirfaces (rcSv yap rmnKC^v r& /ack k(m
wph% ypafjLfjLOii (rvFurra/uicva, ra Sc irpoc ciri^avciatf ). And, since some lines are
plane (cirivcSoi) and others solid (arcpcoi)— those being plane which are simply
conceived of in a plane («Sv cv mirc&p 6,irkfj 17 vArftm), and those solid the
origin of which is revealed from some section of a solid figure, as the cylin-
drical helix and the conic lines (cos rifs tcvXivSpucff^ IXucoi koI ron^ KWfucQi^
ypafifjLwv) — I should say (^rfv av) further that, of locus-theorems on lines,
. some give a plane locus and others a solid locus."

Leaving out of sight for the moment the class of loci on surfaces^ we find
that the distinction between plane and solid loci^ or plane and solid lineSy was
similarly understood by Eutocius, who says (Apollonius, ed. Heiberg, 11.
p. 184) that ^^ solid loci have obtained their name from the fact that the lines
used in the solution of problems regarding them have their origin in the
section of solids, for example the sections of the cone and several others.**
Similarly we gather from Pappus that plane loci were straight lines and circles,
and solid loci were conies. Thus he tells us (vii. p. 672,' 20) that Aristaeus
wrote five books of Solid Loci '' supplementary to (literally, continuous with)
the conies"; and, though Hultsch brackets the passage (vii. p. 662, 10 — 15)
which says plainly that plane loci are straight lines and circles, while solid loci
are sections of cones, i.e. parabolas, ellipses and hyperbolas, we have the
exactly conesponding distinction drawn by Pappus (in. p. 54, 7 — 16) between
plane and solid problems^ plane problems being those solved by means of
straight lines and circumferences of circles, and solid problems those solved
by means of one or more of the sections of the cone. But, whereas Proclus

330 BOOK I [l3S

and Eutocius speak of other solid loci besides conies, there is nothing in
Pappus to support the wider application of the term. According to Pappus
(ill. p. ^4, i6— ai) problems which could not be solved by means of straight
lines, curdes, or conies were linear (yftofAfiuci) because they used for their
construction lines having a more complicated and unnatural origin than thoae
mentioned, namely such curves as quadrairias^ conchoids and dssoids.
Similarly, in the passage supposed to be interpolated, linear loci are distin-
guished as those which are neither straight lines nor drdes nor any of the
conic sections (vii. p. 662, 13 — 15). Thus the classification given by Produs
and Eutodus is less predse than that which we find in Piippus; and the
indusion by Proclus of the cylindrical helix among solid loci, on the ground
that it arises from a section of a solid figure, would seem to be, in any case^
due to some misapprehension.

Comparing these passages and the hints in Pappus about loci on suffaoes
(roiroi vpo« iwi^vtUf) with special reference to Euclid's two books under that
title, Hdberg concludes that loci on lines and loci on surfaces in Produs'
explanation are lod which are lines and lod which are sur&ces respectively.
But some qualification is necessary as reeirds Produs' conception of loci am
lineSy because he goes on to say (p. 395, 5), with reference to this proposition,
that, while the locus is a locus on lines and moreover plane^ it is *'the whole
space between the parallels ** which is the locus of the various paralldograms
on the same base proved to be equal in area. Similarly, when he quotes
iiL 31 about the equality of the angles in the same sc|;ment and iii. 31 about
the right angle in a semicircle as cases where a drcumference of a drde
takes the place of a straight line in a plane locus-theorem, he appears to
imply that it is the segment or semicircle as an area which is regarded as the
locus of an infinite number of triangles with the same base and equal vertical
angles, rather than that it is the circumference which is the locus of the angular
points. Likewise he gives the equality of parallelograms inscribed in "the
asymptotes and the hyperbola " as an example of a solid locus-theorem, as if
the area induded between the curve and its asymptotes was regarded as the
locus of the equal parallelograms. However this may be, it is dear that the
locus in the present proposition can only be either (i) a lineAocys of a line^
not a point, or (3) an areaAoais of an area^ not a point or a line ; and we
seem to be thus brought to another and different classification of lod
corresponding to that quoted by Pappus (vii. p. 660, 18 sqq.) from the pre-
liminary exposition given by Apollonius in his Plane Lod, According to this,
lod in general are of three kinds: (i) c^cjcriico^ holding4n^ in which sense
the locus of a point is a point, of a line a Ime,. of a surface a surface, and of a
solid a solid, (2) SicjoSuco^ matting along^ a line being in this sense a locus of a
point, a sur&ce of a line and a solid of a surface, (3) ayaorpo^o^ where a
surface is a locus of a point and a solid of a line. Thus the locus in this
proposition, whether it is the space between the two parallels regarded as the
locus of the equal parallelograms, or the line parallel to the base regarded as
the locus of the sides opposite to the base, would seem to be of the first dass
(c^cftrucck) ; and, as Proclus takes the former view of it, a locus on lines is
apparently not merely a locus which is a line but a locus bounded by Hnes
idso, the locus bdng plane in the particular case because it is bounded by
straight lines, or, in the case of in. 31, 31, by straight lines and drdes, but
not by any higher curves.

Produs notes lastly (p. 395, 13 — 21) that, according to Geminus,
^'Chrysippus likened locus-theorems to the idecu. For, as the ideas confine

I. 35. 36] PROPOSITIONS 35, 36 . 33^

the genesis of unlimited (particulars) within defined limits, so in such theorems

the unlimited (particular figures) are confined within defined plcues or loci

^ (roiroi). And it is this boundary which is the cause of the equality ; for the

\ height of the parallels, which remains the same, while an infinite number of

i ^ paiullelograms are conceived on the same base, is what makes them all equal

to one another."

Proposition 36.

Parallelograms which are on equal bases and in the same
parallels are equal to one another.

Let ABCD, EFGH be parallelograms which are on
equal bases BC^ FG and in the same parallels AH, BG \

I say that the parallelogram ABCD is equal to EFGH.

For let BEy CH be joined.

Then, since BC is equal to FG,
while FG is equal to EH,

BC is also equal to EH. \C. N. i]

But they are also parallel.

And EB, HC join them ;
but straight lines joining equal and parallel straight lines (at
the extremities which are) in the same directions (respectively)
are equal and parallel. [i. 33]

Therefore EBCH is a parallelogram. [i. 34]

And it is equal to ABCD ;
for it has the same base BC with it, and is in the same
parallels BC, AH with it. [i. 35]

For the same reason also EFGH is equal to the same
EBCH, [1. 35]

so that the parallelogram ABCD is also equal to EFGH.

\C. N. 1]

Therefore etc.

Q. E. D.

L

332 BOOK I [1.37

Proposition 37.

Triangles which are an the same 6ase and in the same
parallels are equal to one another.

Let ABC, DBC be triangles on the same base EC and
in the same parallels AD, BC ;
5 I say that the triangle ABC is equal to the triangle DBC.

Let AD be produced in both
directions lo E, F\

through B let BE be drawn parallel
to CA, [i. 31]

10 and through C let CF be drawn
parallel to ED. [i. 31]

Then each of the figures
EEC A, DECF is a parallelogram ;
and they are equal,
15 for they are on the same base BC and in the same
parallels EC, EF. [i. 3S]

Moreover the triangle ABC is half of the parallelogram
EEC A ; for the diameter AB bisects it [i. 34]

And the triangle DEC is half of the parallelogram DBCF\
10 for the diameter DC bisects it. [i. 34]

[But the halves of equal things are equal to one another.]
Therefore the triangle ABC is equal to the triangle DEC
Therefore etc.

Q. E, D.

31. Here and in the next proposition Heibeig brackets the words **Bat the halves of
eooal things are equal to one another*' on the ground that, since the Common l/otiom
which asserted this fact was interpolated at a very early date (before the time of Theon),
it is probable that the words here were interpolated at the same time. Ct note above
(p. 334) on the interpolated Common Notion,

There is a lacuna in the text of Proclus' notes to i. 36 and i. 37.
Apparently the end of the former and the beginning of the latter are missing,
the Mss. and the editio princeps showing no separate note for i. 37 and no
lacuna, but goin^ straight on without regard to sense. Proclus had evidently
remarked again m the missing passage that, in the case of both parallelograms
and triangles between the same parallels, the two sides which stretch from one
parallel to the other may increase in length to any extent, while the area
remains the same. Thus the perimeter in parallelograms or triangles is of
itself no criterion as to their area. Misconception on this subject was rife
among non-mathematicians; and Proclus (p. 403, 5 sqq.) tells us (i) of
describers of countries (xcapoypo^) who drew conclusions regarding the size
of cities from their perimeters, and (2) of certain members of communistic

(i

I. 37. 38] PROPOSITIONS 37, 38 333

societies in his own time who cheated their fellow members by giving them
land of greater perimeter but less area than they took themselves, so that, on
the one hand, they ^ot a reputation for greater honesty while, on the other, they
took more than their share of produce. Cantor (Gesch, d. Math. i„ p. 172)
quotes several remarks of ancient authors which show the prevalence of the
same misconception. Thus Thucydides estimates the size of Sicily according
to the time required for circumnavigating it About 130 b.c Polybius said
that there were people who could not understand that camps of the same
periphery might have different capacities. Quintilian has a similar remark,
and Cantor thinks he may have had in his mind the calculations of Pliny, who
compares the size of different parts of the earth by adding their length to their
breadth.

The comparison however of the areas of different figures of equal contour
had not been neglected by mathematicians. Theon of Alexandria, in his
commentary on Book i. of Ptolemy's Syn/axiSy has preserved a number of
propositions on the subject taken from a treatise by Zenodorus vcpl UrofUrfHoy
irxqfiaTtay (reproduced in Latin on pp. 11 90 — 121 1 of Hultsch's edition of
Pappus) which was written at some date between, say, 200 B.C. and 90 A.D.,
and probably not long after the former date. Pappus too has at the banning
of Book V. of his ColUction (pp. 308 sqq.) the same propositions, in which he
appears to have followed Zenodorus pretty closely while making some changes
in detail. The propositions proved by Zenodorus and Pappus include the
following: (i) that, of all polygons of the same number of sides and equal
perimeter^ the equilateral and equiangular polygon is the greatest in area^
(2) that, of regular polygons of equal perimeter^ that is the greatest in area
which has the most angles^ (3) that a circle is greater than any regular polygon
of equal contour^ (4) that, of all circular segments in which the arcs are equal in
lengthy the semicircle is the greatest The treatise of Zenodorus was not con-
fined to propositions about plane figures, but gave also the theorem that, of
all solid figures the surfaces of which are equals the sphere is the greatest in
volume.

Proposition 38.

Triangles which are on equal bases and in the same
parallels are equal to one another.

Let ABC, DEF be triangles on equal bases BC, EF and
in the same par^tllels BF, AD ;

I say that the triangle ABC is q a d h

equal to the triangle DEF.

For let AD be produced in
both directions to G, H\
through B let BG be drawn
parallel to CA, [i. 31]

and through F let FH be drawn parallel to DE.

Then each of the figures GBCA, DEFH is a parallelo-
gram ;
and GBCA is equal to DEFH\

334 BOOK I [1.38

for they are on equal bases BC, EF and in the same
parallels BF, GH. [l 36]

Moreover the triangle ABC is half of the parallelogram
GBCA ; for the diameter AB bisects it [i- 34] .

And the triangle FED is half of the parallelogram DEFH\
for the diameter DF bisects it [i. 34]

[But the halves of eaual things are equal to one another.]

Therefore the triangle ABC is equal to the triangle DEF.

Therefore etc.

Q. £. D.

On this proposition Produs remarks (pp. 405 — 6) that Euclid seems to
him to have given in vi. i one proof including all the four theorems from
I- 35 to I. 38, and that most people had failed to notice this. When Eudid,
he says, proves that triangles and paralldograms of the same altitude have to
one another the same ratio as their bases, he simply proves all these
propositions more generally by the use of proportion ; for of course to be of
the same altitude is equivalent to being in the same parallels. It is true that
VI. I generalises these propositions, but it must be observed that it does not
prove the propositions themsdves, as Proclus seems to imply; they are in fiurt
assumed in order to prove vi. i.

Comparison of areas of triangles of I. 24.

The theorem abready mentioned as given by Produs on i. 24 (pp. ^0—4)
is placed here by Heron, who also enunciates it more dearly (an-Nairld, ed.
Besthom-Heiberg, pp. 155— 161, ed. Curtze, pp. 75 — 8).

If in two triangles two sides of the one be equal to two sides of the other
respectively^ and the angle of the one be greater than the angle of the other^
namely the angles contained by the equal sides^ then, (i) if the sum of the two
angles contain^ by the equal sides is equal to two right angles, the two triangles
are equal to one another ; (2) if less than two right angles, the triangle which
has the greater angle is also itself greater than the other; (3) if greater than two
right angles, the triangle which has the less angle is greater them the other
trian^.

D

Let two triangles ABC, DEF have the sides AB, AC respectively equal
to DE, DF.

(i) First, suppose that the angles at A and D in the triangles ABC^
DEFzx^ together equal to two right angles.

Heron's construction is now as follows.

Make the angle EDG equal to the angle BAC

Draw -^-ff' parallel to ED meeting DG in H.

Join Elf.

1.38] PROPOSITION 38 335

Then, since the angles BAC^ EDF zxt equal to two right angles, the
angles EDH^ EDF^x^ equal to two right angles.

But so are the angles EDH^ DHF,

Therefore the angles EDF^ DHFzx^ equal

And the alternate angles EDF^ DFHzit equal. [i. 29]

Therefore the angles DHF, DFHdje equal,

and DF'is equal to DH. [i. 6]

Hence the two sides ED^ DHzx^ equal to the two sides BA^ AC; and
the included angles are equal.

Therefore the triangles ABC, JDEIfaie equal in all respects.

And the triangles £>EFf DEIf between the same parallels are equal.

[i- 37]

Therefore the triangles ABC, DEFzxe equal.

[Proclus takes the construction of Eucl. i. 24, i.e., he makes DH equal to
DFznd then proves that EDy FHzxe parallel]

(2) Suppose the angles BAC, EDFXogeiSxei less than two right angles.

As before, make the angle EDG equal to the angle BAC, draw FH
parallel to ED, and join EH.

In this case the angles EDH, EDF are together less than two right
angles, while the angles EDH, DHFzxe equal to two right angles. [i. 29]

Hence the angle EDF, and therefore the angle DFH, is less than the
angle DHF,

Therefore DH\& less than DF. [i. 19]

Produce DHXq G so that DG is equal to DFox AC, and join EG,

Then the triangle JDEG, which is equal to the triangle ABC, is greater
than the triangle DEH, and therefore greater than the triangle DEF.

(3) Suppose the angles BAC, EDF together greater than two right
angles.

A D

We make the same construction in this case, and we prove in like manner
that the angle DHF\& less than the angle DFH,

whence DH\s greater than DFox AC,

Make DG equal to AC, and join EG.

It then follows that the triangle DEF is greater than the triangle ABC

[In the second and third cases again Proclus starts from the construction
in I. 24, and proves, in the second case, that the parallel, Fff, to ED cuts
DG and, in the third case, that it cuts DG produced.]

336

BOOK I

[i- 3«» 39

There is no necessity for Heron to take account of the position of ^ in
relation to the side opposite D. For in the first and third cases F must fidl

in the position in which Euclid draws it in i. 24, whatever be the relative
lengths of AB^ AC. In the second case the figure may be as aiuiexed, but the
proof is the same, or rather the case needs no proof at all.

Proposition 39.

Equal triangles which are on the same base and an the
same side are also in the same parallels.

Let ABC, DBC be equal triangles which are on the same
base BC and on the same side of it ;
5 [I say that they are also in the same parallels.]
And [For] let AD be joined ;
I say that AD is parallel to BC.

For, if not, let AE be drawn through
the point A parallel to the straight line
^''BC, [1. 31]

and let EC be joined.

Therefore the triangle ABC is equal
to the triangle EBC\

for it is on the same base BC with it and
15 parallels.

But ABC is equal to DBC ;
therefore DBC is also equal to EBC,

the greater to the less : which is impossible.
Therefore AE is not parallel to BC.
Similarly we can prove that neither is any other straight
line except AD ;

therefore AD is parallel to BC.
Therefore etc.

in

the

same

[i- 37]

[C. N. I]

ao

^ 39. 40] PROPOSITIONS 38—40 337

'" 5. [I say that they are also in the same parallels.] Heiberp has proved (Hermes^
3CXXVIII., 1003, p. 50) from a recently discovered papyrus-fragment (Ayum Uwns and their
papyri^ p. 96, No. IX.) that these words are an interpolation by some one who did not observe
that the words **And let AD be joined*' are part of the setting-^ut (Mwtt), bat took them
as belonging to the construction (iraroj'Kein)) and conseauently thought that a dioptfffUs or
"definition '^ (of the thine to be proved) should preceae. The interpolator then altered
"And" into "For" in the next sentence.

This theorem is of course the partial converse of i. 37. In i. 37 we have
triangles which are (i) on the same base, (2) in the same parallels, and the
theorem proves (3) that the triangles are equal. Here the hypothesis (i) and
the conclusion (3) are combined as hypotheses, and the conclusion is the
hypothesis (2) of i. 37, that the triangles are in the same parallels. The
additional qualification in this proposition that the triangles must be on the
same side of the base is necessary because it is not, as in i. 37, involved in the
other hypotheses.

Proclus (p. 407, 4 — 17) remarks that Euclid only converts i. 37 and i. 38
^ relative to triangles, and omits the converses of i. 35, 36 about parallelograms
as unnecessary because it is easy to see that the method would be the same,
and therefore the reader may properly be left to prove them for himself.

The proof is, as Proclus points out (p. 408, 5 — 21), equally easy on the
supposition that the assumed parallel AE meets BD or CD produced
beyond D,

[Proposition 40.

Equal triangles which are on equal bases and on the same
side are also in the same parallels.

Let ABC, CDE be equal triangles on equal bases BC,
CE and on the same side.

I say that they are also in the same parallels.

For let AD be joined ;
I say that AD is parallel to BE. t^ p

For, if not, let AF be drawn throucfh P\ " — y)'^
A parallel to BE [i. 31], and let FE be
joined.

Therefore the triangle ABC is equal
to the triangle FCE ;

for they are on equal bases BC, CE and in the same parallels
BE.AF. [1.38]

But the triangle ABC is equal to the triangle DCE \
therefore the triangle DCE is also equal to the triangle
FCE, _ [C.N.i]

the greater to the less : which is impossible.
Therefore AF is not parallel to BE.

338 BOOK I [1-40^41

Similarly we can prove that neither is any other straight
line except AD ;

therefore AD is parallel to BE.
Therefore etc.

Q. E. D.]

Heiberg has proved b^ means of the papyrus-fragment mentioned in the
last note that this proposition is an interpolation by some one who thou^^t
that there should be a proposition following i. 39 and related to it in the same
way as i. 38 is related to i. 37, and i. 36 to i. 35.

Proposition 41.

If a parallelogram liave the sante base with a triangle and
be in the same parallels, the parallelogram is double of the
triangle.

For let the parallelogram ABCD have the same base EC
with the triangle EBC, and let it be in the same parallels
BC, AE\

I say that the parallelogram ABCD is double of the
triangle BEC.

For let -r4C be joined.

Then the triangle ABC is equal to
the triangle EBC ;

for it is on the same base BC with it
and in the same parallels BC, AE.

[i. 37]

But the parallelogram ABCD is double of the triangle
ABC\

for the diameter AC bisects it ; [i. 34]

so that the parallelogram ABCD is also double of the triangle
EBC.

Therefore etc.

Q. E. D.

On this proposition Proclus (pp. 414, 15 — 415, 16), **by way of practice"
(yv/uturuic Ivcica), considers the area of a trapezium (a quadrilateral with only
one pair of opposite sides parallel) in comparison with that of the triangles
in the same parallels and having the greater and less of the parallel sides of
the trapezium for bases respectively, and proves that the trapezium is less
than double of the former triangle and more than double of the latter. ,
He next (pp. 415, 22 — 416, 14) proves the proposition that,
If a triangle be formed by joining the middle point of either of the non-
parallel sides to the extremities of the opposite side^ the area of the trapezium is
always double of that of the triangle.

1.41,4a] PROPOSITIONS 40— 4a ii9

Let ABCD be a trapezium in which AD^ BC are the ptrallel ades, and
E the middle point of one of the non-parallel sides,
say/7C

Join EA^ EB and produce BE to meet AD
produced in F.

Then the triangles BEC, FED have two angles
equal respectively, and one side CE equal to one
sideZ>£;

therefore the triangles are equal in all respects. [i. 26]

Add to each the quadrilateral ABED ;
therefore the traperium ABCD is equal to the triangle ABF^

that is, to twice the triangle AEB^ since BE is equal to EF. [i. 38]

The three properties proved by Proclus may be combined in one enuncia-
tion thus :

If a triangle be formed by joining the middle point of one side of a trapezium
to the extremities of the opposite side^ the area of the trapezium « (i) greater
than, (2} equal to^ or (3) less than^ double the area of the triangle according as
the side the nUdMe point of which is taken is (i) the greater of the parallel sides^
(2) either of the non-parallel sides^ or (3) the lesser rf the parallel sides.

Proposition 42,

To construct^ in a given rectilineal angle, a parallelogram
equal to a given triangle.

Let ABC be the given triangle, and D the given recti-
lineal angle ;

thus it is required to construct in the rectilineal angle D a
parallelogram equal to the
triangle ABC.

Let BC be bisected at E,
and let AE be joined ;
on the straight line EC, and
at the point E on it, let the
angle CEF be constructed
equal to the angle D ; [i. 23]

through A \et AGhe drawn parallel to EC, and [i. 31]

through C let CG be drawn parallel to EE.

Then FECG is a parallelogram.

And, since BE is equal to EC,

the triangle ABE is also equal to the triangle AEC,
for they are on equal bases BE, EC and in the same parallels
BC,AG\ [1.38]

therefore the triangle ABC is double of the triangle
AEC

22 — 2

340 BOOK I ti-4tM3

But the parallelogram FECG is also double of the triangle
A EC, for it has the same base with it and is in the same
parallels with it ; [i. 41]

therefore the parallelogram FECG is equal to the
triangle ABC.
And it has the angle C^-F equal to the given angle D.
Therefore the parallelogram FECG has been constructed
equal to the given triangle ABC, in the angle C£*/^ which is
equal to D. Q. E. F.

Proposition 43.

In any parallelogram the complements of the parallelograms
about the diameter are equal to one another.

Let A BCD be a parallelogram, and AC\X& diameter ;
and about AC let EH, FG be parallelograms, and BK, KD
5 the so-called complements ;

I say that the complement BK is equal to the complement
KD.

For, since A BCD is a parallelogram, and y4C its diameter,

the triangle ABC is equal to
10 the triangle A CD. [i. 34]

Again, since EH is a parallelo-
gram, and AK is its diameter,

the triangle AEK is equal to
the triangle AHK.
15 For the same reason

the triangle KFC is also equal to KGC.
Now, since the triangle AEK is equal to the trianele
AHK,

and KFC to KGC,
ao the triangle AEK together with KGC is equal to the triangle
AHK together with KFC. \C. N. 2]

And the whole triangle ABC is also equal to the whole
ADC\

therefore the complement BK which remains is equal to the
as complement KD which remains. \C. N. 3]

Therefore eta

Q. E. D.

I- 43» 44] PROPOSITIONS 42—44 34i

I. complements, vapav\fipi&/iaru, the fieures put in to fill up (interstices).

4. and about AC... Euclid's phraseoTofinr here and in the next proposition implies
that the complements as well as the other panulelograms are *' about ** me diagonal. The
words are hete rcpi M Hjr AP vapaXKiiXiypa/ifjia |Uv Irrc* rd B6, ZH, r& N \ey6fi€»a
wtifav\fifn&fMra r& BE, KA. The expression " the so-called complements " indicates that
this technical use of wapawkiipiifiaTa was not new, though it might not be universally known.

In the text of Proclus' commentary as we have it, the end of the note on
I. 41, the whole of that on i. 42, and the beginning of that on i. 43 are
missing.

Proclus remarks (p. 418, 15 — 20) that Euclid did not need to give a
formal definition of campUment because the name was simply suggested by the
facts; when once we have the two "parallelograms about the diameter/'
the complements are necessarily the areas remain-
ing over on each side of the diameter, which fill
up the complete parallelogram. Thus (p. 417,
I sqq.) the complements need not be parallelo-
grams. They are so if the two "parallelograms
about the diameter" are formed by straight lines
drawn through one point of the diameter parallel
to the sides of the original parallelogram, but not

otherwise. If, as in the first of the accompanying figures, the parallelograms
have no common point, the complements are five-sided figures as shown.
When the parallelograms overlap, as in the second figure, Proclus r^ards
the complements as being the small parallelo-
grams FGt EH, But, if complements are strictly
the areas required to fill up the original parallelo-
gram, Proclus is inaccurate in describing Fd EH
as the complements. The complements are really
\ ( I ) the parallelogram FG minus the triangle LMN^
and (2) the parallelogram EH minus the triangle
KMN^ respectively; the possibility that the re-
spective differences may be negative merely means the possibility that the
sum of the two parallelograms about the diameter may be together greater
than the original parallelogram.

In all the cases it is easy to show, as Proclus does, that the complements
are still equal

Proposition 44,

,To a given straight line to apply ^ in a given rectilineal
angle, a parallelogram equal to a given triangle.

Let AB be the given straight line, C the given triangle
and D the given rectUineal angle ;
5 thus it is required to apply to the given straight line AB, in
an angle equal to the angle D, a parallelogram equal to the
I given triangle C.

Let the parallelogram BEFG be constructed equal to

the triangle c, in the angle EBG which is equal to D [i. 42] ;

{lo let it be placed so that BE is in a straight line with AB \ let

I

343 BOOK I [l44

FG be drawn through to H^ and let AH be drawn through
A parallel to either BG or EF. [i. 31]

Let HB be joined.

Then, since the straight line HF falls upon the parallels
isAH.EF,

the angles AHF, HFE are equal to two right angles.

[I. »9]
Therefore the angles BHG^ GFE are less than two right
angles;

and straight lines produced indefinitely from angles less than
«> two right angles meet ; [Post 5]

therefore HB, FE, when produced, will meet
Let them be produced and meet at K\ through the point
K let KL be drawn parallel to either EA or FH, [i. 31]

and let HA, GB be produced to the points L, M.
*s Then HLKF is a parallelogram,
HK is its diameter, and AG, ME are parallelograms, and
LB, BFlht, so-called complements, about HK\

therefore LB is equal to BF. [i. 43]

But BF is equal to the triangle C ;
30 therefore LB is also equal to C [C N. i]

And, since the angle GBE is equal to the angle ABM,

[I. IS]
while the angle GBE is equal to D,

the angle ABM is also equal to the angle D.
Therefore the parallelogram LB equal to the given triangle
35 C has been applied to the given straight line aS, in the angle
ABM which is equal to D.

Q. E. F.

14. since the straight line HP falls.... The verb is in the aorist (^^rf^cr) here and
in similar expreisions in the following propositions.

This proposition will always remain one of the most impressive in all
geometry when accomit is taken (i) of the great importance of the result

i

1. 44] PROPOSITION 44 343

obtained, the transformation of a parallelogram of any shape into another
with the same angle and of equal area but with one side of any given
length, e.g. a unit len^, and (2) of the simplicity of the means employed,
namely the mere apphcation of the property that the complements of the
** parallelograms about the diameter" of a parallelogram are equal. The
marvellous ingenuity of the solution is indeed worthy of the "godlike men of
old," as Proclus calls the discoverers of the method of "application of areas";
and there would seem to be no reason to doubt that the particular solution,
I like the whole theory, was Pythagorean, and not a new solution due to Euclid
tj himself.

Application of areas.

On this proposition Proclus gives (pp. 419, 15 — 420, 23) a valuable note
on the method of " application of areas " here introduced, which was one of
the most powerful methods on which Greek geometry relied. The note runs
as follows :

" These things, says Eudemus (oc vcpc ror £v8i;/ioy), are ancient and are
discoveries of the Muse of the Pythagoreans, I mean the application of areas
(wufiofiokij Tw x^^\ their exceeding (vrcpjSoAi;) and their feUling-short
(IXXci^is). It was from the Pythagoreans that later geometers fi-e. Apollonius]
took the names, which they again transferred to the so-called conic lines,
designating one of these a parabola (application), another a hyperbola
(exceeding) and another an ellipse (falling-short), whereas those godlike men
of old saw the things signified by these names in the construction, in a plane,
of areas upon a finite straight line. For, when you have a straight line set
out and lay the given area exactly alongside the whole of the straight line, then
they say that you apply (TOfMtjSoXXciv) the said area; when however you
make the length of the area greater than the straight line itself, it is said to
excetd (vrcpjSaAXciv), and when you make it less, in which case, after the area
has been drawn, there is some part of the straight line extending beyond it,
it is said to fall short (IkXdvwi). Euclid too, m the sixth book, speaks in
this way both of exceeding 2xA falling-short \ but in this place he needed the
application simply, as he sought to appljr to a given straight line an area equal
to a given triangle in order that we might have in our power, not only the
construction (owrao-i^) of a parallelogram equal to a given triangle, but also
the application of it to a finite straight line. For example, given a triangle
with an area of 12 feet, and a straight line set out the length of which is
4 feet, we apply to the straight line the area equal to the triangle if we take
the whole length of 4 feet and find how many feet the breadth must be in
order that the parallelogram may be equal to the triangle. In the particular
case, if we find a breadth of 3 feet and multiply the length into the breadth,
supposing that the angle set out is a right angle, we shall have the area. Such
then is the application handed down from early times by the Pjrthagoreans."
-^w Other passages to a similar effect are quoted from Plutarch, (i) " Pytha-
goras sacrificed an ox on the strength of his proposition (SuCypofifMi) as
ApoUodotus (?-rus) says... whether it was the theorem of the hypotenuse, viz.
that the square on it is equal to the squares on the sides containing the
right angle, or the problem about the application of an area,*' {Non posu
sucnriter vivi secundum Epicurum^ c. 11.) (2) "Among the most geometrical
theorems, or rather problems, is the following : given two -figures, to apply a
third equal to the one and similar to the other, on the strength of which
discoveiy they say moreover that Pjrthagoras sacrificed. TUs is indeed
unquestionably more subtle and more scientific than the theorem wluch

344 BOOK I [l44

demonstrated that the square on the hypotenuse is equal to the squares on
the sides about the right angle" {Symp. vin. 3, 4).

The story of the sacrifice must (as noted by Bretschneider and Hankd)
be ^ven up as inconsistent with Pythagorean ritual, which forbade sudi
sacnfices ; but there is no reason to doubt that the first distinct formulation
and introduction into Greek geometry of the method of appUcaium af anas
was due to the Pythagoreans. The complete exposition of the appHcatum of
areas, their exceeding and their faUmg-shcrt^ and of the construction of a
rectilineal figure equal to one given figure and similar to another, takes us
into the sixth Book of Euclid ; but it will be convenient to note here the
general features of the theory of application^ exceeding zxAftUUng-skari.

The simple application of a parallelogram of given area to a given
straight line as one of its sides is what we ^ve in i. 44 and 4S ; the g^iend
form of the problem with regard to exceeding zxAfaUimg'Shori may be stated
thus:

"To apply to a given straight line a rectangle (or, more generally, a
parallelogram) equal to a given rectilineal figure and (i) exceeding or
(3) falUngshort by a square (or, in the more general case, a parallelogram
similar to a given parallelogram )•"

What is meant by sayuig that the applied parallelogram (i) exceeds or
(3) faUs short is that, while its base coincides and is coterminous ai one end
with the straight line, the said base (i) overlaps or (3) fidls short of the
straight line at the other end^ and the portion by which the applied
parallelognun exceeds a parallelogram of the same angle and height on the
given straight line (exactly) as base is a parallelogram similar to a given
parallelogram (or, in particular cases, a square). In the case where the
parallelo^am is to fall shorty a Siopur/ioi is necessary to express the condition
of possibility of solution.

We shall have occasion to see, when we come to the relative propositions
in the second and sixth Books, that the general problem here stated is
equivalent to that of solving geometrically a mixed quadratic equation. We
shall see that, even by means of 11. 5 and 6, we can solve geometrically the
equations

s?'ax^V\

but in VI. 38, 39 Euclid gives the equivalent of the solution of the general
equations

b . C
^ c m

We are now in a position to understand the application of the terms

parabola (application), hyperbola (exceeding) and ellipse (falling-short) to

conic sections. These names were first so applied by ApoUonius as expressing

in each case the fundamental property of the curves as stated by him. This

fundamental property is the geometncal equivalent of the (Cartesian equation

referred to any diameter of the conic and the tangent at its extremity as (in

general, oblique) axes. If the parameter of the ordinates fix>m the sevoal

points of the conic drawn to the given diameter be denoted by / (/ being

a^
accordingly, in the case of the hyperbola and ellipse, equal to -j , where d is

a

the length of the given diameter and d* that of its conjugate^ ApoUonius gives

the properties of the three conies in the following form.

I- 44, 4S] PROPOSITIONS 44, 4S 34S

(i) For iht parabola^ the square on the ordinate at any point is equal to
a rectangle applied to / as base with altitude equal to the corresponding
abscissa. That is to say, with the usual notation,

y^px.

(2) For the hyperbola and eilipse^ the square on the ordinate is equal to
the rectangle applied to / having as its width the abscissa and exceeding (for
the hyperbola) ox falling-short (for the ellipse) by a figure similar and similarly
situated to the rectangle contained by the given diameter and p.

x^
That is, in the hyperbola y* =px + 35 pd^

a

or y=/^+3^;

a
and in the ellipse y =/:c "d^'

The form of these equations will be seen to be exactly the same as that of
the general equations above given, and thus Apollonius' nomenclature followed
exactly the traditional theory of application y exceeding^ hnA falling-short.

Proposition 45,

To construct, in a given rectilineal angle, a parallelogram
equal to a given rectilineal figure.

Let A BCD be the given rectilineal figure and E the given
rectilineal angle ;
5 thus it is required to construct, in the given angle E^ a
parallelogram equal to the rectilineal figure ABCD.

o

^^P-L

Let DB be joined, and let the parallelogram FH be

constructed equal to the triangle ABD, in the angle HKF

which is equal to E \ [i. 42]

©let the parallelogram GM equal to the triangle DBC be

applied to the straight line GH, in the angle GrlM which is

^ equal to E. [i. 44]

Then, since the angle E is equal to each of the angles

HKF, GHM,

15 the angle HKF is also equal to the angle GHM. [C N. i]

\

I

346 BOOK I . [i. 45

Let the angle KHG be added to each ;
therefore the angles FKH, KHG are equal to the angles
KHG, GHM.

But the angles FKH, KHG are equal to two right angles;

ao therefore the angles KHG, GHM ^^ also equal to two right
angles.

Thus, with a straight line GH, and at the point H on it,

two straight lines Kff, HM not lying on the same side make

the adjacent angles equal to two right angles ;

25 therefore KH is in a straight line with HM. [l 14]

And, since the straight line HG falls upon the parallels

KM, FG, the alternate angles MHG, HGF are equal to one

another. [l 39]

Let the angle HGL be added to each ;

30 therefore the angles MHG, HGL are equal to the angles

HGF, HGL. [C N. 2]

But the angles MHG, HGL are equal to two right angles ;

[L29]

therefore the angles HGF, HGL are also equal to two right

angles. [C N. i]

35 Therefore FG is in a straight line with GL. [i. 14]

And, since FK is equal and parallel to HG, [i. 34]

and HG to ML also,

KF is also equal and parallel to ML ; [C. ^ i ; i. 30]

and the straight lines KM, FL join them (at their extremities);

40 therefore Km, FL are also equal and parallel. [i. 33]

Therefore KFLM is a parallelogram.

And, since the triangle ABD is equal to the parallelogram

^^' and DBC to GM,

45 the whole rectilineal figure ABCD is equal to the whole
parallelogram KFLM.

Therefore the parallelogram KFLM has been constructed
equal to the given rectilineal figure ABCD, in the angle FKM
which is equal to the given angle E. q. e. f.

St 3f ^% 45* 48* rectilineal figure, in the Greek '* rectilineal " simply, without **fipiie,"
€^^faikiii» being here used as a substantive, like the similarly formed waptXkiKAyptLiitum,

Transformation of areas.

We can now take stock of how Ceu- the propositions i. 43 — 45 bring us in
the matter of transfarmoHon af areas, which constitutes so important a part of

I

1. 45. 46] PROPOSITIONS 4S» 46 347

what has been fitly called the geometrical algebra of fhe Greeks. We have
now learnt how to represent any rectilineal area, which can of course be
resolved into triangles, by a single parallelogram having one side equal to any
^ven straight line and one angle equal to any given rectilineal angle. Most
important of all such parallelograms is the rectangle, which is one of the simplest
forms in which an area can be shown. Since a rectangle corresponds to the
product of two magnitudes in algebra, we see that application to a given
straight line of a rectangle equal to a given area is the geometrical equivalent
of a^ebraical division of the product of two quantities by a third. Further
than this, it enables us to add or subtract any rectilineal areas and to represent
the sum or difference by om rectangle with one side of any given length, the
process being the equivalent of obtaining a common factor. But one step
still remains, the finding of a square equal to a given rectangle, Le. to a
giveii rectilineal figure; and this step is not taken till 11. 14. In general,
die transformation of combinations of rectangles and squares into other
combinations of rectangles and squares is the subject-matter of Book 11., with
the exception of the e)^ression of the sum of two squares as a single square
which appears earlier in the other Pythagorean theorem i. 47. Thus the
transformation of rectilineal areas is made complete in one direction^ i.e. in the
direction of their simplest expression in terms of rectangles and squares, by the
end of Book 11. The reverse process of transforming the simpler rectangular
area into an equal area which shall be similar to any rectilineal figure requires,
of course, the use of proportions, and therefore does not appear till vi. 25.

Proclus adds to his note on this proposition the remark (pp. 422, 24 —
423, 6): "I conceive that it was in consequence of this problem that the
ancient geometers were led to investigate the squaring of the circle as well.
For, if a parallelogram can be found equal to any rectilineal figure, it is worth
inquiring whether it be not also possible to prove rectilineal figures equal to
circular. And Archimedes actually proved that any circle is equal to the
right-angled triangle which has one of its sides about the right angle [the
perpendicular] equal to the radius of the circle and its base equal to the
perimeter of the circle. But of this elsewhere."

Proposition 46.
On a given straight line to describe a square.
Let AB be the given straight line ;
thus it IS required to describe a square
on the straight line AB.
5 Let AC he drawn at right angles to
the straight line AB from the point A
on It [i. 11], and let AD be made equal
to AB ;

through the point Z? let DE be drawn
10 parallel to AB,
and through the point B let B£ be drawn parallel to AD.

348 BOOK I [1.46

Therefore ADEB is a parallelogram ; ^

therefore AB is equal to DE, and AD to BE. [l 34]
But AB is equal to AD ;
IS therefore the four straight lines BA, AD, DE, EB

are equal to one another ;
therefore the parallelogram ADEB is equilateral.
I say next that it is also right-angled.
For, since the straight line AD falls upon the parallels
^AB, DE,

the angles BAD, ADE are equal to two right angles.

[i. 29]
But the angle BAD is right ;

therefore the angle ADE is also right
And in parallelogrammic areas the opposite sides and
15 angles are equal to one another ; [l 34]

therefore each of the opposite angles ABE, BED is also
right

Therefore ADEB is right-angled.
And it was also proved equilateral,
p Therefore it is a square; and it is described on the straight
line AB.

Q. E. F.

f , 3, 30. Produs (p. 4)3, 18 soq.) notes the differenoe between the woid ctmsirmci
(vwHiaaffBai) applied uj Euclid to tne conitniction of a triamgle (and, he might hare added,
of an an^) ana the words describi on (dMi7pd0ccr dT6) used of drawing a square on a given
strai^t line as one side. The triangle (or angU) is, so to say, pieced together, while the
describing of a square on a given strau;ht line is the making of a fipire "from" «if side,
and cotTttpoods to the multiplication ofthe number representmg the side by itself.

Produs (pp. 434 — s) proves that, tf squares art dacribtd an equal shrai^
lines, the squares are equal; and, conversely, that,
if two squares are equal, the straiM lines are
equal on which they are described. The first
proposition is immediately obvious if we divide
the squares into two triangles b]^ drawing a
diagoiial in each. The converse is proved as
follows.

Place the two equal squares AF, CG so
that AB, BC are in a straight line. Then,
since the andes are right, FB, BG will also
be in a straight line. Join AF, FC, CG, GA.

Now, since the squares are equal, the
triangles ABF, CBG are equal

Add to each the triangle FBC-, thmfore the triangles AFC, GFC are
equal, and hence they must be in the same parallels.

1. 46, 47]

PROPOSITIONS 46, 47

349

Therefore AG, CFaxe parallel.

Also, since each of the alternate angles A/^G, FGC is half a right angle,
AF, CG are parallel.

Hence AFCG is a parallelogram ; and AF, CG are equal.

Thus the triangles ABF^ CBG have two angles and one side respectively
equal;
therefore AB is equal to BC^ and BFxo BG.

Proposition 47.

In right-angled triangles the square on the side subtending
the right angle is equal to the squares on the sides containing
the right angle.

Let ABC be a right-angled triangle having the angle
^BAC right;

I say that the square on BC is equal to the squares on
BA, AC.

For let there be described

on BC the square BDEC^

ID and on BA, AC the squares

GB,HC\ [1.46]

through A let AL be drawn

parallel to either BD or CE,

and let AD, FC be joined.

IS Then, since each of the

angles BAC, BAG is right,

it follows that with a straight

line BA, and at the point A

on it, the two straight lines

so/^C AG not lying on the

same side make the adjacent

i angles equal to two right

angles ;

therefore C4 is in a straight line with AG.
For the same reason

BA is also in a straight line with AH.
And, since the angle DBC is equal to the angle FBA : for
each is right :
let the angle ABC be added to each ;

therefore the whole angle DBA is equal to the whole
angle FBC. [C. N. a]

[I. 14]

K5

3SO BOOK I [l47

And, since DB is equal to BC^ and FB to BA^
the two sides AB, BD are equal to the two sides FB^ BC
respectively ;
J5 and the angle ABD is equal to the angle FBC ;
therefore the base AD is equal to the base FC,
and the triangle ABD is equal to the triangle FBC. [l 4]
Now the paraJlelogram BL is double of the triangle ABD,
for they have the same base BD and are in the same parallels
10 ^Z?, AL. [1.41]

And the square GB is double of the triangle FBC,
for they again have the same base FB and are in the same
parallels FB, GC [i. 41]

[But the doubles of equals are equal to one another.]
^5 Therefore the parallelogram BL is also equal to the

square GB.
Similarly, if AE, BK be joined,
the parallelogram CL can also be proved equal to the square
HC\
;o therefore the whole square BDEC is equal to the two

squares GB, HC [C. N. 2] j

And the square BDEC is described on BC, \

and the squares GB, HC on BA, AC
Therefore the square on the side BC is equal to the
15 squares on the sides BA, AC

Therefore etc. q. e. d.

I. die square on, rh dro...rrrpd7a»yo9, the word draTpo^ or dfayrypa/iftlrMr being
understood.

subtending the right angle. Here 6Torecro<^0iff, ** subtending," is used with the
simple accusative (rfyy 6p(9V 7w^) instead of being followed by wh and the accusatiTe,
which seems to be the original and more orthodox construction. Cf. i. i8, note.

33. the two sides AB, BD.... Euclid actually writes ^' DB, BA,^^ and therefore the
equal sides in the two triangles are not mentioned in corresponding order, though he adheres
to the words ixaHpa ixaripq. *' respectively." Here DB is equal to BC and BA to FB.

44. [But the doubles of equals are equal to one another.] Heiberg brackets
these words as an interpolation, since it quotes a Common Notion which is itself interpolated.
Cf. notes on l. 371 p. 331, and on interpolated Common Notions^ pp. )S3 — ^4.

"If we listen," says Proclus (p. 426, 6 sqq.), "to those who wish to
recount ancient history, we may find some of them referring this theorem to
Pythagoras and saying that he sacrificed an ox in honour of his discovery.
But for my part, while I admire those who first observed the truth of this
theorem, I marvel more at the writer of the Elements, not only because he
made it fast (KorcSifcraro) by a most lucid demonstration, but because he
compelled assent to the still more general theorem by the irrefingable
arguments of science in the sixth Book. For in that Book he proves
generally that, in right-angled triangles, the figure on the side subtending
die right angle is equal to the similar and similarly situated figures described
on the sides about me right angle."

1.47] PFOPOSITION 47 iSi

In addition, Plutarch (in the passages quoted above in the note on i. 44),
Diogenes Laertius (viii. 12) and Athenaeus (x. 13) agree in attributing this
proposition to Pythagoras. It is easy to point out, as does G. Junge ("Warm
haben die Griechen das Irrationale entdeckt?^^ in Novae Symboltu Joachimiau^
Halle a. S., 1907, pp. 221 — 264), that these are late witnesses, and that the
Greek literature which we possess belonging to the first five centuries after
Pythagoras contains no statement specifying this or any other particular great
geometrical discovery as due to him. Yet the distich of ApoUodorus the
" calculator,'' whose date (though it cannot be fixed) is at least earlier than
that of Plutarch and presumably of Cicero, is quite definite as to the existence
of one " famous proposition " discovered by Pythagoras, whatever it was. Nor
does Cicero, in commenting apparently on the verses (De not, deor, in. c 36,
§ 88), seem to dispute the fact of the geometrical discovery, but only the story
of the sacrifice. Junge naturally emphasises the apparent uncertainty in the
statements of Plutarch and Proclus. But, as I read the passages of Plutarch,
I see nothing in them inconsistent with the supposition that Plutarch
unhesitatingly accepted as discoveries of Pythagoras both the theorem of the
square of the hypotenuse and the problem of the application of an area, and
the only doubt he felt was as to which of the two discoveries was the more
appropriate occasion for the supposed sacrifice. There is also other evidence
not without bearing on the question. The theorem is closely connected with
the whole of the matter of Eucl. Book 11., in which one of the most prominent
features is the use of the gnomon. Now the gnomon was a well-understood
term with the Pythagoreans (cf. the fragment of Philolaus quoted on p. 141 of
Boeckh's Philolaos des Pythagpreers Lehreti^ 1819). Aristotle also (Physics
III. 4, 203 a 10 — 15) clearly attributes to the Pythagoreans the placing of odd
numbers as gnomons round successive squares beginning with i, thereby
forming new squares, while in another place (Categ, 14, 15 a 30) the word
gnomon occurs in the same (obviously familiar) sense : " e.g. a square, when a
gnomon is placed round it, is increased in size but is not altered in form." The
inference must therefore be that practically the whole doctrine of Book 11. is
Pythagorean. Again Heron (ist cent, a.d.), like Proclus, credits Pythagoras
with a general rule for forming right-angled triangles with rational whole
numbers for sides. . Lastly, the Eudemian summary in Proclus (p. 65, 19)
credits Pythagoras with the discovery of the theory of irrationals. [It is true
that Junge will not accept this either. But in order to support his thesis he
has to reject Friedlein's reading oXoytov (" irrationals ") in favour of avoKoytav
("proportionals"^ or waXoymv (** proportions "), the only basis of which is a
note "(alii ayaXoyuM')" in August's Euclid i. p. 290, and which is certainly
not confirmed by the two Scholia, x. No. i definitely attributing the discovery
of the incommensurable to the Pythagoreans and v. No. i crediting Eudoxus
with the whole theory of Book v. and making no mention of the Pydiagoreans
in this connexion.] Now everything goes to show that this discovery of the
irrational was made with reference to Jz^ the ratio of the diagonal of a
square to its side. It is clear that this presupposes the knowledge that i. 47
is true of an isos^celes right-angled triangle ; and the fact that some triangles
of which it had been discovered to be true were rational right-angled triangles
, was doubtless what suggested the inquiry whether the ratio between the
lengths of the diagonal and the side of a square could also be expressed in
whole numbers. On the whole, therefore, I see no sufficient reason to question
the tradition that, so far as Greek geometry is concerned (the possible priority
of the discovery of Uie same proposition in India will be considered later),

3S«

BOOK I

[1.47

Pythagoras was the first to introduce the theorem of i. 47 and to give a
general proof of it

On this assumption, how was Pythagoras led to this discovery? All that
we can say is that it is probable that the Egyptians were aware that a triangle
with its sides in the ratio 3, 4, 5 was right-angled. Cantor inferred this him
the fact that this was precisely the triangle with which Pythagoras began, if
we may accept the testimony of "N^truvius (ix« 2) that Pythagoras taught how
to make a right angle by means of three lengths measured by the numbers
3> 49 5- If then he took from the Egyptians the triangle 3« 49 Si be presum-
ably learnt its property from them also. Now the Egyptians must certainly
be credited from a period at least as Ceu- back as 2000 &c with the knowledge
that 4* + 3*=S*. This has been proved recently by new evidence. Cantor
(ArMv der Mathefnatik und PhysiA^ viii., 1905, p. 66) refers to a fragment
of papyrus belonging to the time of the 12th Dynasty newly discov^ed at
Kahun. In this papyrus we have extractions of square roots : e.g. that of 16
is 4, that of lyV is ii> that of 6\ is 2^, and the following equations can be
traced:

8«+ 6« =io«

2« + (li)«=(2i)«
l6*+ 12* =20^.

It will be seen that 4* + 3*= 5* can be derived from each of these by
multiplying, or dividing out, by one and the same factor. No doubt
4* + 3'=^' itself was omitted as too well known to need mention. ^ The
Babylonians were, as well as the Egyptians, probably aware that the triangle
with sides 3, 4, 5 was right-angled (Cantor, Gtschichte der Maihematik i,,
pp. 49, 50); the Chinese were certainly aware of it {ibid, p. 181).

How then did Pythagoras discover the general theorem ? Observing that
3, 4, 5 was a right-angled triangle, while 3' + 4' = 5', he was probably led to

consider whether a similar relation was true of the sides of right-anded
triangles other than the particular one. The simplest case (geometrically) to
investigate was that of the isosceles right-angled triangle ; and the truth of the
theorem in this particular case would easily appear from the mere construction
of a figure. Cantor (i„ p. 185) and Allman (Greek Geometry from TTiales to
Eudid^ p. 29) illustrate by a figure in which the squares are drawn outwards,
as in I. 47, and divided by diagonals into equal triangles ; but I think that the
truth was more Jikely to be first observed from a figure of the kind suggested
by Biirk (Das Apastamba'&ulba'Siitra in Zeitschrift der deutschen margenldnd.
Gesellschaft^ lv., 1901, p. 557) to explain how the Indians arrived at the
same thing. The two figures are as shown above. When the geometrical

1. 473 PROPOSITION 47 353

consideration of the figure had shown that the isosceles right-angled triangle
had the property in question, the investigation of the same fact from the
arithmetical point of view would ultimately lead Pythagoras to the other
momentous discovery of the irrationality of the length of the diagonal of a
square expressed in terms of its side.

The irrational will come up for discussion later ; and our next question
is: Assuming that Pythagoras had observed the geometrical truth of the
theorem in the case of the two particular triangles, and doubtless of other
rational right-angled triangles, how did he establish it generally ?

There is no positive evidence on this point. Two possible lines are
however marked out. (i) Tannery says (La Ghmktrie grecgue^ p. 105) that
the geometry of Pythagoras was sufficiently advanced to make it possible
for Um to prove the theorem by similar triangles. He does not say in
what particular manner similar triangles would be used, but their use must
apparently have involved the use of proportions^ and, in order that the proof
should be conclusive, of the theory of proportions in its complete form
applicable to incommensurable as well as commensurable magnitudes. Now
Eudoxus was the first to make the theory of proportion independent of the
hypothesis of commensurability ; and as, before Eudoxus* time, this had not
been done, any proof of the general theorem by means of proportions given
by Pythagoras must at least have been inconclusive. But this does not
constitute any objection to the supposition that the truth of the general
theorem may have been discovered in such a manner ; on the contrary, the
supposition that Pythagoras proved it by means of an imperfect theory of
proportions would better than anything else account for the fact that Euclid
had to devise an entirely new proof, as Proclus says he did in i. 47. This
proof had to be independent of the theory of proportion even in its rigorous
form, because the plan of the Elements postponed that theory to Books v.
and VI., while the Pythagorean theorem was required as early as Book 11.
On the other hand, if the Pythagorean proof had been based on the doctrine
of Books I. and 11. only, it would scarcely have been necessary for Euclid to
supply a new proof.

The possible proofs by means of proportion would seem to be practically
limited to two.

{a) One method is to prove, from the similarity of the triangles ABC^
DBA, that the rectangle CB, BD \% equal to the
square on BA, and, from the similarity of the
triangles ABC, DAC, that the rectangle BC, CD
is equal to the square on CA \ whence the result
follows by addition.

It will be observed that this proof is in substance
identical with that of Euclid, the only difference
being that the equality of the two smaller squares
to the respective rectangles is inferred by the method of Book vi. instead
of from the relation between the areas of parallelograms and triangles on the
same base and between the same parallels established in Book i. It occurred
to me whether, if Pythagoras^ proof had come, even in substance, so near to
Euclid's, Proclus would have emphasised so much as he does the originality
of Euclid's, or would have gone so far as to say that he marvelled more at
that proof than at the original discovery of the theorem. But on the whole
I see no difficulty ; for there can be little doubt that the proof by proportion
is what suggested to Euclid the method of i. 47, and the transformation of

H. E. 23

354

BOOK I

[1.47

the method of proportions into one based on Book i. only, effected by a
construction and proof so extraordinarily ingenious, is a veritable i&tir de
farce which compels admiration, notwithstanding the ignorant strictures of
Schopenhauer, who wanted something as obvious as die second figure in
the case of die isosceles right-angled triangle (p. 352), and accordingly
(Sammtliche Werke^ 111. § ^9 and i. § 15) calls Eudid's proof *'a mouse-trap
proof" and "a proof walku^ on stilts, nay, a mean, underhand, proof (^'Des
Eukleides stelzbeiniger, jja, mnter listiger Beweis **).

{b) The other possible method is this. As it would be seen that the
trian^es into which the original triangle is divided by the perpendicukr from
the nght angle on the hypotenuse are similar to one another and to the whole
trian^e, while in these three triangles the t^*o sides about the right angle in die
original triangle, and the hypotenuse of the original triangle, are corr^ponding
sides, and that the sum of the two former similar triangles is identiodly equal
to the similar triangle on the hypotenuse, it might be inferred that the same
would also be true of squares described on tihe corresponding three sides
respectively, because squares as well as similar triangles are to one another in
the duplicate ratio of corresponding sides. But the same thing is equally true
of any similar rectilineal figures, so that this proof would practically establish
the extended theorem of Eucl. vi. 31, which theorem, however, Produs
appears to regard as being entirely Euclid's discovery.

On the whole, the most probable supposition seems to me to be that
Pythagoras used the first method (a) of proof by means of the theory of
proportion as he knew it, i.e. in the defective form which was in use up to the
date of Eudoxus.

(2) I have pointed out the difficulty in the way of the suj^xisition that
Pythagoras' proof depended upon the principles of EucL Books i. and 11. only.

. c 1— ^

a

\

\

Were it not for this difficulty, the conjecture of Bretschneider (p. 82^, followed
by Hankel (p. 98), would be the most tempting hypothesis. According to this
suggestion, we are to suppose a figure like that of Eucl. 11. 4 in which a, ^ are
the sides of the two inner squares respectively, and a + ^ is the side of the
complete square. Then, if the two complements, which are equal, are divided
by their two diagonals into four equal triangles of sides a, ^, r, we can place
these triangles round another square of the same size as the whole square, in the
manner shown in the second figure, so that the sides a, h of successive triangles
make up one of the sides of the square and are arranged in cyclic order. It
readily follows that the remainder of the square when the four triangles are
deducted is, in the one case, a square whose side is ^, and in the other the sum of
two squares whose sides are a, ^ respectively. Therefore the square on c is equal

47]

PROPOSITION 47

355

to the sum of the squares on a, b. All that can be said against this con-
jectural proof is that it has no specifically Greek colouring
but rather recalls the Indian method. Thus Bhaskara
(bom 1 114 A.D. ; see Cantor, !«, p. 656) simply draws
four right-angled triangles equal to the original one in-
wards, one on each side of the square on the hypotenuse,
and says "see!", without even adding that inspection
shows that

^ = 4- + (a-iJ)* = a" + ^.

Though, for the reason given, there is difficulty in supposing that
Pythagoras used a general proof of this kind, which applies of course to right-
angled triangles with sides incommensurable as well as commensurable, there
is no objection, I think, to supposing that the truth of the proposition in the
case of the first rational right-angled triangles discovered, e.g. 3, 4, 5, was
proved by a method of this sort. Where the sides are commensurable in this
way, the squares can be divided up into small (unit) squares, which would
much facilitate the comparison between them. That this subdivision was in
fact resorted to in adding and subtracting squares is made probable by
Aristotle's allusion to odd numbers as gnomons placed round unity to form
successive squares in Physics 111. 4; this must mean that the squares were
represented by dots arranged in the form of a square and a gnomon formed of
dots put round, or that (if the given square was drawn in the usual way) the
gnomon was divided up into unit squares. Zeuthen has shown (" Thhrhne
de jythagore" Origine di la Giomttrie scientifique in Comptes rendus du
II"^ Congrh international de Philosophiey Geneve, 1904), how easily the
^position could be proved by a method of this kind for the triangle 3, 4, 5.
To admit of the two smaller squares being shown side by side, take a square
on a line containing 7 units of length (4 + 3), and divide it up into 49
small squares. It would be obvious that the
whole square could be exhibited as containing p

four rectangles of sides 4, 3 cyclically arranged
round the figure with one unit square in the
middle. (This same figure is given by Cantor, i„
p. 680, to illustrate the method given in the
Chinese " Tcheou pei ".) It would be seen that

(i) the whole square (7*) is made up of two
squares 3' and 4*, and two rectangles 3, 4;

(ii) the same square is made up of the square
EFGH and the halves of four of the same rect-
angles 3, 4, whence the square EFGH^ being equal

to the sum of the squares 3* and 4*, must contain 2^ unit squares and its side,
or the diagonal of one of die rectangles, must contam 5 units of length.

Or the result might equally be seen by observing that

(i) the square EFGH on the diagonal of one of the rectangles is made
up of the halves of four rectangles and the unit square in the middle, while

(ii) the squares 3* and 4' placed at adjacent comers of the large square
make up two rectangles 3, 4 with the unit square in the middle.

The procedure would be equally easy for any rational right-angled triangle,
and would be a natural method of trying to praise the property when it had

23—2

/

/

V

/

y

*"

\

A

\

"l

\

\

\

\

k^

\

\

/■

y

\

^

35^

BOOK I

[1.47

once been empirically observed that triangles like 3> 4* 5 did in fact contain a
right angle.

2^utiben has, in the same paper, shown in a most ingenious way how the
property of the triangle 3, 4, 5 could be verified by a sort of combination of
the second possible method by similar triangles, ,
{b) on p. 354 above, with subdivision of rectangles
into similar small recian^s. I give the method on
account of its interest, although it is no doubt too
advanced to have been used by those who first
proved the property of the particular triangle.

Let ABC be a triangle right-angled at A^ and
such that the lengths of the sides AB^ACzxt^ and
3 units respectively.

Draw the perpendicular AD^ divide up AB^ AC
into unit lengths, complete the rectangle on BC as

base and with AD as altitude, and subdivide this rectangle into small
rectangles by drawing parallels to BCy AD through the points of division of
AB.AC

Now, since the diagonals of the small rectangles are all equal, each being
of unit length, it follows by similar triangles that the small rectangles are all
equal. And the rectangle with AB for diagonal contains 16 of the small
rectangles, while the rectangle with diagonal A C contains 9 of them.

But the sum of the triangles ABD^ ADC is equal to the triangle ABC.

Hence the rectangle with BC as diagonal contains 9 + 16 or 35 of the
small rectangles ;

and therefore BC= 5.

Rational right-angled triangles from the arithmetical stand-
point.

Pythagoras investigated the arithmetical problem of finding rational
numbers which could be made the sides of right-angled triangles, or of finding
square numbers which are the sum of two squares ; and herein we find the
beginning of the indeterminate analysis which reached so high a stage of
development in Diophantus. Fortunately Proclus has preserved Pythagoras'
method of solution in the following passage (pp. 428, 7 — 429, 8). ''Certain
methods for the discovery of triangles of this kind are hand^ down, one of
which they refer to Plato, and another to Pythagoras. [The latter] starts from
odd numbers. For it makes the odd number the smaller of the sides about
the right angle; then it takes the square of it, subtracts unity, and makes
half the difference the greater of the sides about the right angle; lastly it adds
unity to this and so forms the remaining side, the hypotenuse. For example,
taking 3, squaring it, and subtracting unity from the 9, the method takes naif
of the 8, namely 4 ; then, adding unity to it again, it makes 5, and a right-
angled triangle has been found with one side 3, another 4 and another 5. But
the method of Plato argues from even numbers. For it takes the given even
number and makes it one of the sides about the right angle ; then, bisecting
this number and squaring the half, it adds unity to the square to form the
hypotenuse, and subtracts unity from the square to form the other side about
the right angle. For example, taking 4, the method squares half of this, or
2, and so makes 4 ; then, subtracting unity, it produces 3, and adding unity
it produces 5. Thus it has formed the same triangle as that which was
obtained by the other method."

1.47] PROPOSITION 47 ' 357

The formula of Pythagoras amounts, if m be an odd number, to

the sides of the right-angled triangle being «, , . Cantor

(ij, pp. 185 — 6), taking up an idea of Roth {Geschichte der abmdHindischen
Phdosophie^ ii. 5^.7)1 ^ves the following as a possible explanation of the way in
which Pythagoras arrived at his formula. If ^ = a' + ^', it follows that

a» = ^-^ = (r + ^)(r-^).

Numbers can be found satisfying the first equation if (i) r+ ^ and r — ^ are
either both even or both odd, and if further (2) ^+^ and c-b are such
numbers as, when multiplied together, produce a square number. The first
condition is necessary because, in order that c and b may both be whole
numbers, the sum and difference oi c + b and c-b must both be even. The
second condition is satisfied i( c-k-b and r-^ are what were called similar
numbers (^/aococ AptOfioC) ; and that such numbers were most probably known
in the time before Plato may be inferred from their appearing in Theon of
Smyrna {Expasitio rerum mathematicarum ad iegendum Platonem utilium^ ed.
Hiller, p. 36, 12), who says that similar plane numbers are, first, all square
numbers and, secondly, such oblong numbers as have the sides which contain
them proportional Thus 6 is an oblong number with length 3 and breadth 2 ;
24 is another with length 6 and breadth 4. Since therefore 6 is to 3 as 4 is
to 2, the numbers 6 and 24 are similar.

Now the simplest case of two similar numbers is that of i and a*, and,
since i is odd, the condition (i) requires that a*, and therefore a^ is also odd.
That is, we may take i and (2/1+1)* and equate them respectively to ^-^ and
c-^b^ whence we have

(2;fn)«>-i
^" 2

c = ^ + 1 ,

while a = 2/f + I.

As Cantor remarks, the form in which c and b appear correspond sufficiently
closely to the description in the text of Proclus.

Another obvious possibility would be, instead of equating c-bx,o unity, to
put ^-^=2, in which case the similar number c-^b must be equated to
double of some square, i.e. to a number of the form 211*, or to the half of an

even square number, say ^ — '- . This would give

2

a= 211,
■ r = «*+ I,

which is Plato's solution, as given by Proclus.

The two solutions supplement each other. It is interesting to observe that
the method suggested by Roth and Cantor is very like that of Eucl x.
(Lemma i following Prop. 28). We shall come to this later, but it may be
mentioned here that the problem is to find two square numbers such that their

358 BOOK I [1.47

sum is also a square. Euclid there uses the property of 11. 6 to the effect that,
if AB is bisected at C and produced to Z>,

AD.DB + Ba^Ciy.
We may write this uv = ^-l^^

where u^c-^b^ v^c-^b.

In order that uv may be a square, Euclid points out that u and v must be
similar numbers, and further that u and v must be either both odd or both
even in order that b may be a whole number. We may then put for the
similar numbers, sa^, a^S* and a/, whence (if o\^, ay' are either both odd or
both even) we obtain the solution

■^■■^^{^^'f^'-

But I think a serious, and even fatal, objection to the conjecture of Cantor
and Roth is the very fact that the method enables both the Pythagoreajn and
the Platonic series of triangles to be deduced with equal ease. If this had
been the case with the method used by Pythagoras, it would not, I think, have
been left to Plato to discover the second series of such triangles. It seems to
me therefore that Pythagoras must have used some mediod which would
produce his rule aniy ; and further it would be some less recondite method,
su^ested by direct observation rather than by argument from general
pnnciples.

One solution satisfying these conditions is that of Bretschndder (p. 83),
who suggests the following simple method. Pythagoras was certainly aware
that the successive odd numbers are gnomons^ or the differences between
successive square numbers. It was then a simple matter to write down in
three rows (a) the natural numbers, {p) their squares, (r) the successive odd
numbers constituting the differences between the successive squares in {p\ thus :
12345 6 7 8 9 10 II 12 13 14
I 4 9 16 25 36 49 64 81 100 121 144 169 196
I 3 5 7 9 " 13 15 17 19 21 23 25 27
Pythagoras had then only to pick out the numbers in the third row which are
squares, and his rule would be obtained by finding the formula connecting the
square in the third line with the two adjacent squares in the second line. But
even this would require some little argument; and I think a still better
suggestion, because making pure observation play a greater part, is that of
P. Treutlein {Zeiischrift fur Mathematik und Physik, xxviii., 1883, Hist-litt
Abtheilung, pp. 209 sqq.).

We have the best evidence (e.g. in Theon of Smyrna) of the practice of
representing square numbers and other figured numbers, e.g. oblong, triangular,
hexagonal, by dots or signs arranged in the shape of the particular figure.
(Cf. Aristotle, Metaph, 1092 b 12). Thus, says Treutlein, it would be easily
seen that any square number can be turned into the next higher square
by putting a single row of dots round two adjacent sides, in the form of a
gnomon (see figures on next page).

If a is the side of a particular square, the gnomon round it is shown by
simple inspection to contain 2a + i dots or imits. Now, in order that 2a + i
may itself be a square, let us suppose

2<f + I = «*,
whence a = J («* - i),

and a + I = J (^ + i).

1

«-47]

PROPOSITION 47

359

In order that a and a + i may be integral, // must be odd, and we have at
once the Pythagorean formula

I think Treutlein's hypothesis is shown to be the correct one by the passage
in Aristotle's Physics already quoted, where the reference is undoubtedly to the
Pythagoreans, and odd numbers are clearly identified with gnomons ''placed
round i." But the ancient commentaries on the passage make the matter
clearer still. Philoponus says: "As a proof... the Pythagoreans refer to what

12

^ K

J'

^18

happens with the addition of numbers; for when the odd numbers are
successively added to a square number they keep it square and equilateral....
Odd numbers are accordingly called gnomons because, when added to what are
already squares, they preserve the square form.... Alexander has excellently
said in explanation that the phrase ' when gnomons are placed round ' means
making a figure with the cndd numbers (1^ icara rov9 w-cpcrrovf apctffAovs
(rxi7fuiroypa^v)...for it is the practice with the Pythagoreans to represent
things in figures (crxiyftaToypa^ccv)."

The next question is: assuming this explanation of the Pythagorean
formula, what are we to say of the origTn of Plato's ? It could of course be
obtained as a particular case of the general formula of Eucl. x. already
referred to; but there are two simple alternative explanations in this case also,
(i) Bretschneider observes that, to obtain Plato's formula, we have only to
double the sides of the squares in the Pythagorean formula,
for (2/1)' + («•-!)* = («•+ i)*,

where however n is not necessarily odd.

(2) Treutlein would explain by means of an extension of the gnomon idea.
As, he says, the Pythagorean formula was obtained by placing a gnomon
consisting of a single row of dots round two adjacent sides of a square, it
would be natural to try whether another solution could not
be found by placing round the square a gnomon consisting of ' • • •

a double row of dots. Such a gnomon would e(}ually turn the [ * j ] * *
square into a larger square; and the question would be I I|I I I
whether the double-row gnomon itself could be a square. If ••!•••
tlie side of the original s<juare was n, it would easily be seen
that the number of units m the double-row gnomon would be 4^7 + 4, and we
have only to put

4<? + 4=4/1*,

36o BOOK I [1.47

whence a = ff-i,

a-i- 2 = «'-i' I,
and we have the Platonic formula

(2«/ + («•-!)•= («Vi)'.

I think this is, in substance, the right explanation, but, in form, not quite
correct The Greeks would not, I think, have
treated the doubie row as a gnomon. Their com-
parison would have been between (i) a certain •••••••

square //tfi a single-row gnomon and (2) the same •••••••

square fftinus a single-row gnomon. As the IIIIIII

application of Eucl. 11. 4 to the case where the • • • I • I •

segments of the side of the square are a^ i enables

the Pythagorean formula to be obtained as •••••••

Treutlein obtains it, so I think that Eucl. 11. 8 *******

confirms the idea that the Platonic formula was '

obtained by comparing a square plus a gnomon

with the same square minus a gnomon. For 11. 8 proves that

whence, substituting i for by we have

4fl + (a-i)* = (a+i)*,
and we have only to put a = n^io obtain Plato's formula.

The "theorem of Pythagoras" in India.

This question has been discussed anew in the last few years as. the result
of the publication of two important papers by Albert Biirk on Das Apastamba-
^liJba-Su/ra in the Zeitschrift der deutschen margenidndischtn Gtselhchfrft
(lv., 1901, pp. 543— 59i» and lvi., 1902, pp. 327—391). The first of
the two papers contains the introduction and the text, the second the
translation with notes. A selection of the most important parts of the
material was made and issued by G. Thibaut in the Journal of the Asiatic
Society 0/ Bengal^ xliv., 1875, Part i. (reprinted also at Calcutta, 1875,
as The Suhasutras^ by G. Thibaut). Thibaut in this work gave a most
Suable comparison of extracts fi-om the three Sulvasutras by B^udhftyana,
Apastamba and K^tyftyana respectively, with a running commentary and an
estimate of the date and ori^nality of the geometry of the Indians. Biirk
has however done good service by making the Apastamba-S.-S. accessible in
its entirety and investigating the whole subject afresh. With the natural
enthusiasm of an editor for the work he is editing, he roundly maintains, not
only that the Pythagorean theorem was known and proved in all its generality
by the Indians long before the date of Pythagoras (about 580 — 500 &cX but 1
that they had also discovered the irrational; and further that, so iax fi-om
Indian geometry being indebted to the Greek, the much-travelled Pythagoras
probably obtained his theory from India (loc, cit. lv., p. 575 note). Three impor- ;
tant notices and criticisms of Biirk's work have followed, by H. G. Zeuthen /
C Theorkme de Pythagpre^^ Origine de la Giomitrie scientifiquey 1904, already
quoted), by Moritz Cantor ( Vher die dlteste indische Mathematik in the Archfy
der Matlumatik und Physiky viii., 1905, pp. 63—72) and by Heinrich Vogt
{Haben die alten Inder den Pythagoreischen Lehrsatz und dcLS IrrationcUe
gekanntf in the Bibliotheca Mathetnaticay vii„ 1906, pp. 6—23. See also
Cantor's Geschichte der Mathematiky i„ pp. 635—645.

1.47] PROPOSITION 47 361

The general eflfect of the criticisms is, I think, to show the necessity for
the greatest caution, to say the least, in accepting Biirk's conclusions.

I proceed to give a short summary of the portions of the contents of the
Apastamba-§.-S. which are important in the present connexion. It may be
premised that the general object of the book is to show how to construct
altars of certain shapes, and to vary the dimensions of altars without altering
the form. It is a collection of rules for carrying out certain constructions.
There are no proofs, the nearest approach to a proof being in the rule for
obtaining the area of an isosceles trapezium, which is done by drawing a
perpendicular from one extremity of the smaller of the two parallel sides to
the greater, and then taking away the triangle so cut off and placing it, the
other side up, adjacent to the other equal side of the trapezium, thereby
transforming the trapezium into a rectangle. It should also be observed that
Apastamba does not speak of right-angled triangles^ but of two adjacent sides
and the diagonal of a rectangle. For brevity, I shall use the expression
" rational rectangle " to denote a rectangle the two sides and the diagonal of
which can be expressed in terms of rational numbers. The references in
brackets are to the chapters and numbers of Apastamba's work.

(i) Constructions of right angles by means of cords of the following
relative lengths respectively:

3)

a, 4)

(2) A general enunciation of the Pythagorean theorem thus: "The
diagonal of a rectangle produces [i.e. the square on the diagonal is ecjual to]
the sum of what the longer and shorter sides separately produce [i.e. the
squares on the two sidesy' (i. 4)

(3) The application of the Pythagorean theorem to a square instead of a
rectangle [i.e. to an isosceles right-angl^ triangle] : '^The diagonal of a square

ft produces an area double [of the original square]." (i. 5)

(4) An approximation to the value of J2 ; the diagonal of a square is

if I + - + ) times the side. (i. 6)*

\ 3 3.4 3-4.34/

I (5) Application of this approximate value to the construction of a square

with side of any length. (11. i)

(6) The construction of a ^3, by means of the Pythagorean theorem, as
the diagonal of a rectangle with sides a and a Ji, (11. 2)

(7) Remarks equivalent to the following :

(a) a n/J is the side of \ {a JzY* or a VJ= Ja ^3. (11. 3)

(b) A square on length of i unit gives i unit square (111. 4)
„ „ 2 units gives 4 unit squares (in. 6)

» 3 I. 9 » 0"-6)

li » ^i » ("I. 8)

3> 4> 5

(»• 3, V

13, 16, 20

(V.3)

»5. 20, 25

(V.3)

5. ". 13

(v.4)

IS, 36. 39

(1. 2, V.

8, IS. 17

(v.s)

".35,37

(v-S)

3^2

BOOK 1

[1-47

A square on length of 2| units gives 6\ unit squares (iii. 8)
„ „ i unit gives \ unit square (in. lo)

,, „ i « V M ("I- lo)

(c) Generally, the square on any length contains as many rows (of
small, unit, squares) as the length contains units. (in. 7)

(8) Constructions, by means of the Pythagorean theorem, of

(a) the sum of two squares as one square, (n. 4)

(d) the difference of two squares as one square. (n. 5)

(9) A transformation of a rectangle into a square. (n. 7)
[This is not directly done as by Euclid in 11. 14, but the rectangle is first

transformed into a gnomon, i.e. into the diflference

between two squares, which difference is then trans- ^ ^

formed into one square by the preceding rule. If

ABCD be the given rectangle of which BC is the

longer side, cut off the square ABEF^ bisect the

rectangle DE left over by HG parallel to FE^ move

the upper half DG and place it on AF as base in the

position AK. Then the rectangle ABCD is equal to

the gnomon which is the difference between the square

LB and the square LF, In other words, Apastamba

transforms the rectangle ab into the difference between

the squares ( j and f J .]

(10) An attempt at a transformation of a square (a^) into a rectangle
which shall have one side of given length (^). (in. i)

[This shows no sign of such a procedure as that of Eucl. 1. 44, and indeed
does no more than say that we must subtract ab from c^ and then adapt the
remainder e^-ab ^o that it may '' fit on " to the rectangle ab. The problem
is therefore only reduced to another of the same kind, and presumably it was
only solved arithmetically in the case where a, b are given numerically. The
Indian was therefore far from the general, geometrical, solution.]

(i i) Increase of a given square into a larger square. (ni. 9)

{This amounts to saying that you must add two rectangles (a, b) and
another square (^) in order to transform a square a* into a square (a + Vf,
The formula is therefore that of Eucl. n. 4, a* + lab + ^* = (a + bf^

The first important question in relation to the above is that of date.
Biirk assigns to the Apastamba-^ulba'Siitra a date at least as early as the 5th
or 4th century b.c. He observes however (what is likely enough) that the
matter of it must have been much older than the book itself. Further, as
regards one of the constructions for right angles, that by means of cords of
lengths 15, 36, 39, he shows that it was known at the time of the Tdittiriya-
Samhita and the Satapatha-Brdhmana^ still older works belonging to the
8th century b.c at latest When however Biirk says {loc, cit, lv., p. 575) that
the theorem of the square of the hypotenuse and rational right-angled
triangles cannot be found anywhere so early as in India, it would appear
that he is mistaken, if the formulae (all obtainable from 4' + 3' = 5* by
multiplying throughout by the squares of integers or fractions) which are
quoted by Cantor from the recently discovered Egyptian papyrus of date more

1-47] PROPOSITION 47 363

than 2000 years b.c. really justifies the assumption that they were known to
represent rational right-angled triangles.

As regards the various '' rational rectangles '' used by Apastamba, it is to
be observed that two of the seven, viz. 8, 15, 17 and 12, 35, 37, do not belong
to the Pythagorean series, the others consist of two which belong to it, viz. 3,
4, 5 and 5, 12, 13, and multiples of these. It is true, as remarked by
Zeuthen {pp. ciL p. 842), that the rules of 11. 7 and in. 9, numbered (9) and
(11) above respectively, would furnish the means of finding any number of
'^rational rectangles.'' But it would not appear that the Indians had been
able to formulate any general rule; otherwise their list of such rectangles
would hardly have been so meagre. Apastamba mentions seven only, really
reducible to four (though one other, 7, 24, 25, appears in the B|udh2yana-
l§.-S., supposed to be older than Apastamba). These are all that Apastamba
knew of, for he adds (v. 6): "So many recognisable (erkennbare) constructions
are there," implying that he knew of no other "rational rectangles" that could
be employed. But the words also imply that the theorem of the square on
the diagonal is also true of other rectangles not of the " recognisable " kind,
ie. rectangles in which the sides and the diagonal are not m the ratio of
integers; this is indeed implied by the constructions for ^2, ^3 etc. up to ^^6
(cf. II. 2, viii. 5). This is all that can be said. The theorem is, it is true,
enunciated as a general proposition, but there is no sign of anything like a
general proof; there is nothing to show that the assumption of its universal
truth was founded on anything better than an imperfect induction from a
certain number of cases, discovered empirically, of triangles with sides in the
ratio of whole numbers in which the property (i) that the square on the
longest side is equal to the sum of the squares on the other two sides was
found to be always accompanied by the property (2) that the latter two sides
include a right angle.

It remains to consider Biirk's claim that the Indians had discovered the
irrational. This is based upon the approximate value of ^2 given by
Apastamba in his rule i. 6 numbered (4; above. There is nothing to show
how this was arrived at, but Thibaut's suggestion certainly seems the best and
most natural. The Indians may have observed that 17'= 289 is nearly
( double of 12'= 144. If so, the next question which would naturally occur to
them would be, by how much the side 1 7 must be diminished in order that
the square on it may be 288 exactly. If, in accordance with the Indian
' fisishion, a gnomon with unit area were to be subtracted from a square with
17 as side, this would approximately be secured by giving the gnomon the
breadth ^, for 2 x 17 x ^^^ = i. The side of the smaller square thus arrived
at would be 17 — ^=12 + 4+1- ^, whence, dividing out by 12, we have

^2 = 1+- + , approximately.

j ^3 3-4 34. 34 ^^ ^ .

But it is a far cry from this calculation of an approximate value to the

i discovery of the irrational. First, we ask, is there any sign that this value

.^ was known to be inexact? It comes directly after the statement (i. 6) that

i the square on the diagonal of a square is double of that square, and the rule is

■ quite boldly stated without any qualification : " lengthen the unit by one-third

■ and the latter by one-quarter of itself less one-thirty-fourth of this part"
I Further, the approximate value is actually used for the purpose of constructing
I a square when the side is given (11. i). So familiar was the formula that it

364 BOOK I I1.47

Thibaut observes {^Journal of the Asiatic Society of Bengal^ xlix., p. 241) that,
according to B^udh^yana, the unit of length was divided into 12 fingerbreadths^
and that one of two divisions of ihtfingerbrecuHh was into 34 sesame^oms^ and
he adds that he has no doubt that this division, which he has not elsewhere
met, owes its origin to the formula for ^2. The result of using this sub-
division would be that, in a square with side equal to 12 fingerbreadths^ the
diagonal would be \i fingerbreadths less i sesame-corn. Is it conceivable that
a subnlivLsion of a measure of length would be based on an evaluation known
to be inexact ? No doubt the first discoverer would be aware that the area of
a gnomon with breadth ^ and outer side 1 7 is not exactly equal to i but less
than it by the square of ^ or by Wtv' ^^^ therefore that, in taking that
gnomon as the proper area to be suotracted from 1 7^ he was leaving out of
account the small fraction tAt> ^ however, the object of the whole
proceeding was purely practical, he would, without hesitation, ignore this as
being of no practiod importance, and, thereafter, the formula would be
handed down and taken as a matter of course without arousing suspicion as
to its accuracy. This supposition is confirmed by reference to the sort of
rules which the Indians allowed themselves to regard as accurate. Thus
Apastamba himself gives a construction for a circle equal in area to a given
square, which is equivalent to taking v = 3-09, and yet observes that it gives the
required circle ** exactly ^^ (iii. 2), while his construction of a square equal to
a circle, which he equally calls ''eicact," makes the side of the square equal
to x|ths of the diameter of the circle (111. 3), and is equivalent to taking
7 = 3.004. But, even if some who used the approximation for J 2 were
conscious that it was not quite accurate (of which there is no evidence), there
is an immeasurable difference between arrival at this consciousness and the
discovery of the irrational. As Vogt says, three stages had to be passed
through before the irrationality of the diagonal of a square was discovered in
any real sense, (i) All values found by direct measurement or calculations
based thereon have to be recognised as being inaccurate. Next (2) must
supervene the conviction that it is impossible to arrive at an accurate arithmetical
expression of the value. And lastly (3) the impossibility must be proved.
Now there is no real evidence that the Indians, at the date in question, had
even reached the first stage, still less the second or third.

The net results then of Biirk's papers and of the criticisms to which they
have given rise appear to be these, (i) It must be admitted that Indian
{[eometry had reached the stage at which we find it in Apastamba quite
mdependentl^ of Greek influence. But (2) the old Indian geometry was
purely empincal and practical, far removed from abstractions such as the
irrational The Indians had indeed, by trial in particular cases, persuaded
themselves of the truth of the Pythagorean theorem and enunciated it in all
its generality ; but they had not established it by scientific proof.

Alternative proofs.

I. The well-known proof of i. 47 obtained by putting two squares side
by side, with their bases continuous, and cutting oflf right-angled triangles
which can then be put on again in different positions, is attributed by
an-Nairiz! to Thftbit b. Qurra (826—901 a.d.).

His actual construction proceeds thus.

Let ABC be the given triangle right-angled at A.

Construct on AB the square AD\
produce AC to Fso that EFmay be equalto AC.

47]

PROPOSITION 47

365

Construct on EF the square EG, and produce Dff to A!" so that DK
may be equal to AC,

It is then proved that, in the triangles
BAC, CFG, KHG, BDK,
the sides BA, CFy KH, BD are all equal,
and
the sides AC, FG, HG, DKzxe all equal.

The angles included by the equal sides
are all right angles ; hence the four triangles
are equal in all respects. [i. 4]

Hence BC, CG, GK, KB are all equal.

Further the angles DBK, ABCdse equal;
hence, if we add to each the angle DBC,
the angle KBC is equal to the angle ABD
and is therefore a right angle.

In the same way the angle CGK is right ;
therefore BCGK is a square, i.e. the square on BC

Now the sum of the quadrilateral GCLH and the triangle LDB together
with two of the equal triangles make the squares on AB, AC, and together
with the other two make the square on BC,

Therefore etc.

II. Another proof is easily arrived at by taking the particular case of
Pappus' more general proposition given below in which the given triangle
is right-angled and the parallelograms on the sides containing the right angles
are squares. If the figure is drawn, it will be seen that, with no more than
one additional line inserted, it contains Thdbit's figure, so that Thibif s proof
may have been practically derived from that of Pappus.

III. The most interesting of the remaining proofs seems to be that
shown in the accompanying figure.
It is given by J. W. Miiller, Systema-
tische ZusammensteUung der wichtigsten
bisher btkannten Beweise des Pythag:^
Lehrsatzes (Niimberff, 1S19), and in
the second edition (Mainz, 1821) of
Ign. Hoffmann, Der Pythag. Lehr-
satz mit 32 thnls btkannten theih
neuen Beweisen [3 more in second
edition]. It appears to come from
one of the scientific papers of IJon-
ardo da Vinci (1452—1519).

The triangle HKL is constructed
on the base KH with the side KL
equal to BC and the side Z^ equal
to AB.
r Then the triangle IfZK is equal in all respects to the triangle ABC,

I and to the triangle EBF.
Now DB, BG, which bisect the angles ABE, CBF respectively, are
in a straight line. Join BL.
It is easily proved that the four quadrilaterals ADGC, EDGE, ABLK,
HLBC are all equal.

366 BOOK I [1.47

Hence the hexagons ADEFGC, ABCHLKtJt equal
Subtracting from the former the two triangles ABQ EBF^ and from the
latter the two equal triangles ABC^ HLK^ we prove that

the square CK is equal to the sum of the squares AE^ CF,

Pappus' extension of I. 47.

In this elegant extension the triangle may be any triangle (not necessarily
right-angled), and any parallelograms take the place of squares on two of the
sides.

Pappus (iv. p. 177) enunciates the theorem as follows :

If ABC b€ a triangie^ and any parallelograms whatever ABED, BCFG
be described an AB, BC, and if DE, FG be
produced to H, and HB be joined^ the
parallelograms ABED, BCFG are equal
to the parallelogram contained by AC,
HB in an angle which is equal to the
sum of the angles BAC, DHB.

Produce HB to K\ through A, C
draw AL^ Cilf parallel to HK^ and join
LM.

Then, since ALHB is a parallelo-
gram, ALy HB are equal and parallel.
Similarly MC^ HB are equal and parallel

Therefore AL^ MC are equal and
parallel;
whence LM^ ACaie also equal and parallel,

and ALMC is a parallelogram.

Further, the angle LAC oK this parallelogram is equal to the sum of the
angles BAC^ DHBy since the angle DHB is equal to the angle LAB.

Now, since the parallelogram DABE is equal to the parallelogram LABH
(for they are on the same base AB and in the same parallels AB^ DH\
and likewise LABHis equal to LAKN (for they are on the same base LA
and in the same parallels LA^ HK)^

the parallelogram DABE is equal to the parallelogram LAKN,

For the same reason,

the parallelogram BGFC is equal to the parallelogram NKCM.

Therefore the sum of the parallelograms DABE^ BGFC is equal to the
parallelogram LACM^ that is, to the parallelogram which is contained by AC^
HB in an angle LAC which is equal to the sum of the angles BAC^ BHD.

"And this is far more general than what is proved in the Elements about
squares in the case of right-angled (triangles).''

Heron's proof that AL, BK, CF in Euclid's figure meet in
a point.

The final words of Proclus' note on 1. 47 (p. 429, 9—15) are historically
interesting. He says: "llie demonstration by the writer of the Elements being
clear, I consider that it is unnecessary to add anything further, and that we may
be satisfied with what has been written, since in fact those who have added
anvthin^ more, like Pappus and Heron, were obliged to draw upon what is
*^- •tvth Book, for no really useful object." These words cannot

\

1.47] PROPOSITION 47 367

of course refer to the extension of 1. 47 given by Pappus ; but the key to
them, so far as Heron is concerned, is to be found in the commentary of
an-NairizI (pp. 175—185, ed. Besthom-Heiber^ ; pp. 78 — 84, ed Curtze) on
1. 47, wherein he gives Heron's proof that the hnes AL^ FQ BK in Euclid's
figure meet in a point Heron proved this by means of three lemmas which
would most naturally be proved from the principle of similitude as laid down
in Book vi., but which Heron, as a tour de force^ proved on the principles of
Book I. only. 'YYit first lemma is to the following effect

l/y ina triangie ABC, DE be draivn parallel to the base BC, and if AF be
drawn from the vertex A to the middle point F of BC, then AF wilt also
bisect DE.

This is proved by drawing IfIC through A parallel p# Q'

to £>£ or BQ and If£>L, /CEM throu^^ D, E re- tv^y- •.--■; d

spectively parallel to AGF^ and lastly joinmg DF^ EF \ ly'''

Then the triangles ABF, AFC are equal (being Hi — -i^ -m

on equal bases), and the triangles DBF^ EFC are also / jr\ \ /

equal (being on equal bases and between the same p//^ JQ \/ c
parallels). /h^V^^

Therefore, by subtraction, the triangles ADF^ AEF Ol F^ M u
are equal, and hence the parallelograms AL^ Ahf bx^
equal.

These parallelograms are between the same parallels Zif, HK\ therefore
Z/; FM are equal, whence DG^ GE are also equal.

The second lemma is an extension of this to the case where DE meets
BA^ CA produced beyond A,

The third lemma proves the converse of Euclid 1. 43, that, If a paral-
lelogram AB is cut into four others ADGE, DF, FGCB, CE, so that DF,
CE are equals the common vertex G will be on the diagonal AB.

Heron produces AG till it meets CF in H. Then, if we join HB^ we
have to prove that AHB is one straight line. The
proof is as follows. Since the areas DF^ EC are
equal, the triangles DGF^ ECG are equal.

If we add to each the triangle GCF^

the triangles ECF^ DCF are equal ;
therefore ED^ CF 2Jt parallel.

Now it follows from i. 34, 29 and 26 that the
triangles AKE^ GKD are equal in all respects;

therefore EK is equal to KD,

Hence, by the second lemma,

CH is equal to HF.

Therefore, in the triangles FHB^ CHG,
the two sides BF^ Fff^xt equal to the two sides GCy CHy

and the angle BFH is equal to the angle GCH\
hence the triangles are equal in all respects,
and the angle BHF\& equal to the angle GHC

Adding to each the angle GHF^ we find that the angles BHF^ FHG are
equal to the angles CHG, GHF,

and therefore to two right angles.

Therefore AHB is a straight line.

368

BOOK I

[i- 47» 4«

Heron now proceeds to prove the proposition that, in the accompanying
figure, if AKL perpendicular to BC meet
EC in M, and if BM, MG be joined,
BM^ MG are in one straight line.

Parallelograms are completed as shown
in the figure, and the diagonals OA^ FH
of the parallelogram FHsjt drawn.

Then the triangles FAJI, BAC are
clearly equal in all respects;

therefore the angle HFA is equal to

the angle ABC^ and therefore to the angle

C^^ (since AK is perpendicular to BC).

But, the diagonals of the rectangle

J^ cutting one another in K,

FY\s equal to YA,

and the angle HFA is equal to the

angle OAF.
Therefore the angles OAF, CAK are
equal, and accordingly

OA, AK are in a straight line.
Hence OM is the diagonal of SQ \

therefore AS is equal to AQ,
and, if we add AM to each,

FM is equal to MH.
But, since EC is the diagonal of the parallelogram FN,

FM is equal to MN.
Therefore MH is equal to MN\
and, by the third lemma, BM, MG are in a straight line.

Proposition 48.

If in a triangle the square on one of the sides be equal to
the squares on the remaining two sides of the triangle, the
angle contained by the remaining two sides of the triangle is
right.

For in the triangle ABC let the square on one side BC
be equal to the squares on the sides BA, AC;

I say that the angle BAC is right.

For let AD be drawn from the point A at
right angles to the straight line AC, let AD
be made equal to BA, and let DC be joined.

Since DA is equal to AB,
the square on DA is also equal to the square
on AB.

Let the square on -^C be added to each ;

uJ

I. 48] PROPOSITIONS 47, 48 369

therefore the squares on DA, AC are equal to the squares
on BAy AC.

But the square on DC is equal to the squares on DA,
AC, for the angle DAC is right ; [1. 47]

and the square on BC is equal to the squares on BA, AC, for
this is the hypothesis ;

therefore the square on DC is equal to the square on BC,

so that the side DC is also equal to BC
And, since DA is equal to AB,
and AC is common,

the two sides DA, AC are equal to the two sides BA,
AC;

I and the base DC is equal to the base BC ;
I therefore the angle DAC is equal to the angle BAC. [i. 8]

i But the angle DAC is right ;
' therefore the angle BAC is also right.

\ Therefore etc. q. e. d

Proclus' note (p. 430) on this proposition, though it does not mention
I Heron's name, gives an alternative proof, which is the same as that definitely
1 attributed by an-NairizI to Heron, the only diflference being that Proclus
demonstrates two cases in full, while Heron dismisses the second with a
r " similarly." The alternative proof is another instance of the use of i. 7 as a
means of answering objections. If, says Proclus, it be not admitted that the
perpendicular AD may be drawn on the opposite side of AC from B, we may
draw it on the same side as AB, in which case it is impossible that it should
not coincide with AB, Proclus takes two cases,
first supposing that the perpendicular falls, as AD,
1 within the angle CAB, and secondly that it falls,
; as A£, outside that angle. In either case the
absurdity results that, on the same straight line A C
and on the same side of it, AD, DC must be re-
spectively equal to AB, BC, which contradicts i. 7.

Much to the same effect is the note of De Morgan that there is here '' an
appearance of avoiding indirect demonstration by drawing the triangles on
diflferent sides of the base and appealing to i. 8, because drawing them on the
same side would make the appeal to i. 7 (on which, however, i. 8 is founded)."

H. K. 24

BOOK II.

DEFINITIONS.

1. Any rectangular parallelogram is said to be contained
by the two straight lines containing the right angle.

2. And in any parallelogrammic area let any one whatever
of the parallelograms about its diameter with the two comple-
ments be called a gnomon.

Definition i.

Ilav wapaXXrikoypafifLov dpOoyJu^wv vtpc^cirAu Xfycrcu vw6 tvo rmt^ r^
fy$^v ytaytav vcpicxovcraiv cMctwF.

As the full expression in Greek for "the angle BAC" is "the angle
contained by the (straight lines) BA^ AC!* V vvo rmv BA, AF wtpt^x^nhmi
ywvio, so the full expression for "the rectangle contained by BA^ AC"
IS TO viro rwv BA, AF vcpicxo/icvov 6p$oytivio¥. In both cases the
substantive and participle can be omitted because the feminine or neuter of
the article enables us to distinguish whether an angle or a rectangle is meant ;
but the difference in Euclid's phraseology is that the words ^ko tmt BA, AF
appear always in full for the rectangle, whereas the shorter ^w6 BAF is used in
describing the angle. Archimedes and Apollonius, on the other hand,
frequently use the expression to vwb BAF for the rtdan^ BA^ A C^ just as
they use 17 viro BAF for the angle BAC.

Definition 2.

IlaKTOf Si v€LpaXkfiXoypafifi€v xmpiov riSv v€pi r^ Sca/icrpor a^rov TopaXXif-
Xoype^ifUtfv tv 6irotOKovv <rw rote fiixrl wapavKtfpiitfJUun yr«^Miv KoXwOm.

Meaning literally a thing enabling something to be knawn^ observed or
verified^ a teiler or marker^ as we might say, the word gnomon (yi^Mir) was
first used in the sense (i) in which it appears in a passage of Herodotus (11. 109)
stating that " the Greeks learnt the toAoc, the gnomon and the twelve parts of
the day from the Babylonians." According to Suidas, it was Anaximander
(611 — 545 B.C) who introduced the gnomon into Greece. Whatever may be
the details of the construction of the two instruments called the WXot and
the gnomon^ so much is certain, that the gnomon had to do with tbe

II. DEF. 2] DEFINITIONS i, a 371

measurement of time by shadows thrown by the sun, and that the word
signified the placing of a staff perpendicular to the horizon. This is borne
out by the statement of Proclus that Oenopides of Chios^ who first investigated
the problem (Eucl. i. 12) of drawing a perpendicular from an external point
to a given straight line, called the perpendicular a straight line drawn
^^ gnomon-wise^^ (iccitA yF«dfioya). Then (2) we find the
term used of a mechanical instrument for drawing right
angles, as shown in the figure annexed. This seems to be
the meaning in Theognis 805, where it is said that the
envoy sent to consult the oracle at Delphi should be
'' straighter (tfvrcpo«) than the ropvof, the ordBfirf and the
gnomon^ and all three words evidently denote appliances,
the Topvog being an instrument for drawing a circle
(probably a string stretched between a fixed and a moving point), and the
cra$firf a plumb-hne. Next (3) it was natural that the gnomon, owing to its
shape, should become the figure which remained of a square when a smaller
square was cut out of one comer (or the figure, as Aristotle says, which when
added to a square increases its size but does not alter its form). We have
seen (note on i. 47, p. 35 1) that the Pythagoreans used the term in this sense, and
further applied it, by analogy, to the series of odd numbers as having the same
property in relation to square numbers. The earliest evidence for this is the
fragment of Philolaus (c, 460 b.c) already mentioned (see Boeckh, Philoiaos
des Pythagoreers Lehren, p. 141) where he says that " number makes all things
knowable and mutually agreeing («-orayopa dXAoXoic) in the way characteristic of
the gnomon " (icara yi^ttftoKoc ^vcriv). As Boeckh says (p. 144), it would appear
from the fragment that the connexion between the gnomon and the square to
which it is added was r^;arded as symbolical of union and agreement, and that
Philolaus used the idea to explain the knowledge of things, making the
knowing embrace and grasp the hiown as the gnomon does the square. Cf.
Scholium 11. No. 11 (Euclid, ed. Heibeig, Vol v. p. 225), which says ''It is
to be noted that the gnomon was discovered by geometers with a view to
brevity, while the name came from its incidental property, namely that from
it the whole is known, whether of the whole area or of the remainder, when it
is either placed roimd or taken awav. In sundials too its sole function is to
make the actual time of da^ known. '

The geometrical meaning of the word is extended in the definition of
gnomon given by Euclid, where (4) the gnomon has
the same relation to any parallelogram as it before
had to a square. From the fact that Euclid says
*' Ut " the figure described '' he called a gnomon " we
may infer that he was using the word in the wider
sense for the first time. Later still (5) we find
Heron of Alexandria ( I St cent A.D.)defining B.gnomon
in general as any figure which, when added to any
figure whatever, makes the whole figure similar to that to which it is added.
In this definition of Heron (Def. '59) Hultsch brackets the words which make it
apply to any number as well ; but Theon of Smyrna, who explains that plane,
triangular, square, solid and other kinds of numbers are so called after the
likeness of the areas which they measure, does make the term in its most
general sense apply to numbers. "All the successive numbers which fby
being successively added] produce triangles or squares or polygons are called
gnomons'* (p. 37, 11 — 13, ed Hiller). Thus the successive odd numbers added

24— «

^

37« BOOK II

together make square numbers; the gnomons in the case of triangular
numbers are the successive numbers i, 2, 3, 4...; those for pentagonal
numbers are the series i, 4, 7, 10... (the common difference being 3), and so
on. In general, the successive gnomonic numbers for any polygonal number,
say of n sides, have n- 2 for their common difference (Theon of Smyrna,
P- 34, 13—15)-

Geometrical Algebra.

We have already seen (cf part of the note on 1. 47 and the above note on
the gnomoft) how the Pythagoreans and later Greek mathematicians exhibited
different kinds of numbers as forming different geometrical figures. Thus,
says Theon of Smyrna (p. 36, 6 — 11), *' plane numbers, triangular, square
and solid numbers, and the rest, are not so called independently (tsvplm) but
in virtue of their similarity to the areas which they measure ; for 4, since it
measures a square area, is called square by adaptation from it, and 6 is called
oblong for the same reason." A " plane number " is similarly described as a
number obtained by multiplying two numbers together, which two numbers
are sometimes spoken of as "sides," sometimes as the "length" and
" breadth " respectively, of the number which is their product

The product of two numbers was thus represented geometrically by the
rectangle contained by the straight lines representing the two numbers
respectively. It only needed the discovery of incommensurable or irrational
straight lines in order to represent geometrically by a rectangle the product of
any two quantities whatever, rational or irrational ; and it was possible to ad-
vance from a geometrical arithmetic to a geometrical algebra^ which indeed by
Euclid's time (and probably long before) had reached such a stage of develop-
ment that it could solve the same problems as our algebra so far as they do
not involve the manipulation of expressions of a degree higher than the
second. In order to make the geometrical algebra so generally effective, the
theory of proportions was esseritial. Thus, suppose that jt, y^ z etc. are
quantities which can be represented by straight lines, while a, )3, y etc are
coefficients which can be expressed by ratios between straight lines. We can
then by means of Book vi. And a single straight line d such that

cuc + )8>' + y« + ... =^.
To solve the simple equation in its general form

CLT + tf = ^,

where a represents any ratio between straight lines, also requires recourse to
the sixth Book, though, e.g., if a is | or ^ or any submultiple of unity, or if a is
2, 4 or any power of 2, we should not require anything beyond Book i. for
solving the equation. Similarly the general form of a qimdratic equation
requires Book vi. for its geometrical solution, though particular quadratic
equations may be so solved by means of Book 11. alone.

Besides enabling us to solve geoiQetrically these particular quadratic
equations. Book 11. gives the geometrical proofs of a number of algebraical
formulae. Thus the first ten propositions give the equivalent of the several
identities

1. a(^ + r + //4 ...) = ab-\-ac-\-ad-¥ ...,

2. {a^-b)a'k-{a-^b)b = (a-\- b)\

3. {a + b)a = ab'i' n*,

4. (/I + ^)* = n* + ^ + 2ab.

GEOMETRICAL ALGEBRA 373

, ^.(i±i.,)-.(iii)',

or(a + i8)(a-i8) + /9» = a»,

6. (2a + /^)^ + a« = (a + ^)«,
or(a + )8)(i8-a) + a»=/?«,

7. (a + ^)* + tf* = 2 (a + ^) <J + ^,
ora« + /9«=2o)8^(o-)8)«,

8. 4(« + ^)tf + ^ = {(« + ^) + «n
or 4ai8 + («-/»)« = (a + i8)»,

or (a + fl)«+ (a-i8)«= 2 (a«4./5»),
10. (2a + ^)« + ^ = 2 {a* + (fl f ^)»},
or (a + Pf + (i8- a)» = 2 (a« + /9»).

The form of these identities may of course be varied according to the different
symbols which we may use to denote particular portions of the lines given in
Euclid's figures. They are, for the most part, simple identities, but there is no
reason to suppose that these were the only applications of the geometrical
algebra that Euclid and his predecessors had been able to make. We may
infer the very contrary from the fact that Apollonius in his Conies frequently
states without proof much more complicated propositions of the kind.

It is important however to bear in mind that the whole procedure of
Book II. is geometrical \ rectangles and squares are shown in the figures, and
the equality of certain combinations to other combinations is proved by those
figures. We gather that this was the classical or standard method of proving
such propositions, and that the algebraical method of proving them, with no
figure except a line with points marked thereon, was a later introduction.
Accordingly Eutocius' method of proving certain lemmas assumed by
Apollonius (Conies^ ii. 23 and 111. 29) probably represents more nearly than
Pappus' proof of the same the point of view from which Apollonius regarded
them.

It would appear that Heron was the first to adopt the algebraical method
' of demonstratmg the propositions of Book 11., be^nning from the second,
without figures, as consequences of the first proposition corresponding to

According to an-NairizI (ed. Curtze, p. 89), Heron explains that it is not
possible to prove 11. i without drawing a number of lines (i.e. without actually
drawing the rectangles), but that the following propositions up to 11. 10
inclusive can be proved by merely drawing one line. He distinguishes two
varieties of the method, one by dissolution the other by composition by which he
seems to mean splitting-up of rectangles and squares, and combination of them
into others. But in his proofs he sometimes combines the two varieties.

When he comes to 11. 11, he says that it is not possible to do without a
figure because the proposition is a problem, which accordingly requires an
operation and therefore the drawing of a figure.

The algebraical method has been preferred to Euclid's by some English
editors ; but it should not find favour with those who wish to preserve the

374 BOOK II

essential features of Greek geometry as presented by its greatest exponents, or
to appreciate their point of view.

It may not be out of place to add a word with reference to the geometrical
equivalent of the algebraical operations. The addition and sumraction of
quantities represented in the geometrical algebra by lines is of course effected
by producing the line to the required extent or cutting off a portion of it The
equivalent of multiplication is the construction of the rectangle of which the
given lines are adjacent sides. The equivalent of the division of one quantity
represented by a Une by another quantity represented by a line is simply the
statement of a ratio between lines on the principles of Books v. and vi. llie
division of a product of two quantities by a third is rq)resented in the
geometrical algebra by the finding of a rectangle with one side of a given
length and equal to a given rectangle or square. This is the problem of
application of areas solved in i. 44, 45. The addition and subtraction of
products is, in the geometrical a4;ebray the addition and subtraction of
rectangles or squares ; the sum or difference can be transformed into a single
rectangle by means of the application of areas to any line of given len^^
corresponding to the algebraical process of finding a common measure. Lastly,
the extraction of the square root is, in the geometrical algebra, the finding of a
square equal to a given rectangle, which is done in 11. 14 with the help of i. 47.

I

BOOK II. PROPOSITIONS.

!

Proposition i.

If there be two straight lines y and one of them be cut into
any number of segments whatever^ the rectangle contained by
the two straight lines is equal to the rectangles contained by the
uncut straight line and each of the segments.

5 Let A, BC be two straight lines, and let BC be cut at
random at the points /?, E ;

I say that the rectangle contained by A, BC is equal to the
rectangle contained by A, BD,
that contained by A, DE and
lo that contained by A^ EC.

For let BF be drawn from B
at right angles to BC ; [i. n]

let BG be made equal to Ay [i. 3]
through G let GH be drawn
,5 parallel to BC, [i. 31]

^ and through D, E, C let DK,
EL, CH be drawn parallel to
BG.

Then BH is equal to BK, DL, EH.
20 Now BH is the rectangle A, BC, for it is contained by
GB, BC, and BG is equal x,o A \

BK is the rectangle A, BD, for it is contained by GB,
BD, and BG is equal to A ;

and DL is the rectangle A, DE, for DK, that is BG [1. 34],
25 is equal to A.

Similarly also EH is the rectangle A, EC.
Therefore the rectangle A, BC is equal to the rectangle
A, BD, the rectangle A, DE and the rectangle A, EC.
Therefore etc.

Q. E. D.

K L H

376

BOOK II

[IL I,

la the rectangle A, BC. From thb point onward I shall translate thos in cases where
Euclid leaves out the word contained {wtpux^tivoif). Though the word "rectangle '* is also
omitted in the Greek (the neuter article being sufficient to show that the rectangle is
meant), h cannot be dispensed with in English. De Morgan advises the use of the expres-
sion *' the rectangle under two lines.** Inis does not seem to me a very pood expression,
and, if used in a translation from the Greek, it might suggest that M m ri ^6 meant
under^ which it does not.

This proposition, the geometrical equivalent of the algebraical formula
<i(^ + r+^+ ...) = fl^ + ar-i-<i^+...,
can, of course, easily be extended so as to correspond to the more general
algebraical proposition that the product of^ an expression consisting of any
number of terms added together and another expression also consisting of
any number of terms added together is equal to the sum of all the products
obtained by multiplying each term of one expression by all the terms of the
other expression, one after another. The geometrical proof of the more
general proposition would be effected by means of a figure showing all the
rectangles corresponding to the partial products, in the same way as they are
shown in the simpler case of ii. i ; the difference would be that a series of
parallels to BC would have to be drawn as well as the series of parallels
to^-^

Proposition 2.

If a straight line be cut at random^ the rectangle contained
by the whole and both of the segments is equal to the square on
the whole.

For let the straight line AB be cut at random at the
point C\

I say that the rectangle contained by AB, BC together with
the rectangle contained by BA, AC is equal
to the square on AB.

For let the square ADEB be described
on AB [i. 46], and let CF be drawn through
C parallel to either AD or BE. [i. 31]

Then AE is equal to AF, CE.

Now AE is the square on AB ;

AF is the rectangle contained by BA,
AC, for it is contained by DA, AC, and
AD is equal to AB ;

and CE is the rectangle AB, BC, for BE is equal to
AB.

Therefore the rectangle BA, AC together with the rect-
angle AB, BC is equal to the square on AB.

Therefore etc.

Q. E. D.

I

II. 2] PROPOSITIONS I, 2 377

The fact asserted in the enunciation of this proposition has already been
used in the proof of i. 47 ; but there was no occasion in that proof to observe
that the two rectangles BI^ CL making up the square on BC are the
rectangles contained by BC and the two parts, respectively, into which it is
divided by the perpendicular from A on BC. It is this fact which it is
necessary to state in this proposition, in accordance with the plan of Book 11.

The second and third propositions are of course particular cases of the
first They were no doubt separately enunciated by Euclid in order that they
might be immediately available for use hereafter, instead of having to be
deduced for the particular occasion from 11. i. For, if they had not been thus
separately stated, it would scarcely have been practicable to quote them later
without explaining at the same time that they are included in 11. i as particular
cases. And, though the propositions are not used by Euclid in the later
propositions of Book 11., they are used afterwards in xiii. 10 and ix. 15
respectively; and they are of extreme importance for geometry generally,
being constantly used by Pappus, for example, who frequently quotes the
third proposition by the Book and number. •

Attention has been called to the fact that 11. i is never used by Euclid ;
and this may seem no less remarkable than the fact that 11. 2, 3 are not again
used in Book 11. But it is important, I think, to observe that the proofs of
all the first ten propositions of Book 11. are practically independent of each
other, though the results are really so interwoven that they can often be
deduced from each other in a variety of ways. What then was Euclid's
intention, first in inserting some propositions not immediately required, and
secondly in making the proofs of the first ten practically independent of
each other? Surely the object was to show the power of the method of
geometrical algebra as much as to arrive at results. From the point of view
of illustrating the method^ there can be no doubt that Euclid's procedure is
far more instructive than the semi-algebraical substitutes which seem to find
a good deal of favour; practically it means that, instead of relying on our
memory of a few standard formulae, we can use the machinery given us by
Euclid's method to prove immediately ab initio any of the propositions taken
at random.

Let us contrast with Euclid's plan the semi-algebraical alternative. One
editor, for example, thinks that, as 11. i is not used by Euclid afterwards, it
seems more logical to deduce from it those of the subsequent propositions
which can be readily so deduced. Putting this idea into practice, he proves
II. 2 and 3 by quoting 11. i, then proves 11. 4 by means of 11. i and 3, 11. 5 and
6 by means of 11. i, 3 and 4, and so on. The result is ultimately to deduce
the whole of the first ten propositions from 11. i, which Euclid does not use at
all; and this is to give an importance to 11. i which is altogether dispro-
portionate and, by starting with such a narrow foundation, to make the whole
structure of Book 11. top-heavy.

Editors have of course been much influenced by a desire to make the
proofs of the propositions of Book 11. easier, as they think, for schoolboys.
But, even from this point of view, is it an improvement to deduce 11. 2 and 3
from II. I as corollaries ? I doubt it For, in the first place, Euclid's figures
visualise the results and so make it easier to grasp their meaning ; the truth
of the propositions is made clear even to the eye. Then, in the matter of
brevity, to which such an exaggerated importance is attached, Euclid's proof
positively has the advantage. Counting a capital letter or a collocation of such
as one word, I find, e.g., that Mr H. M. Taylor's proof of 11. 2 contains

378

BOOK II

["• «. 3

1 20 words, of which 8 represent the construction. Euclid's as above trans-
lated has 126 words, of which 22 are descriptive of the construction; therefore
the actual proof by Euclid has 8 words fewer than Mr Taylor's, and the extra
words due to the construction in Euclid are much more than atoned for by
the advantage of picturing the result in the figure.

The advantages then which Euclid'sf method may claim are, I think, these:
in the case of 11. 2, 3 it produces the result more easily and clearly than does
the alternative proof by means of 11. i, and, in its general application, it is
more powerful m that it makes us independent of any recoUection of results.

/

F O

B

Proposition 3.

If a straight line be cut at random, the rectangle contained
by the whole and one of the segments is equal to the rectangle
contained by the segments and the square on the aforesaid
segment.

For let the straight line AB be cut at random at C ;
I say that the rectangle contained by AB^ BC is equal to the
rectangle contained by AC, CB together
with the square on BC.

For let the square CDEB bie de-
scribed on CB\ [1.46]
let ED be drawn through to /%
and through A let AF be drawn parallel
to either CD or BE. [i. 31]

Then AE is equal to AD, CE.

Now AE is the rectangle contained by AB^ BC, for it is
contained by AB^ BE, ^na BE is equal to BC ;

AD is the rectangle AC, CB, for DC is equal to CB ;

and DB is the square on CB.
Therefore the rectangle contained by AB, BC is equal to
the rectangle contained by AC, CB together with the square
on BC.

Therefore etc

Q. E. D,

If we leave out of account the contents of Book 11. itself and merely look
to the applicability of propositions to general use, this proposition and the
preceding are, as already indicated, of great importance, and particularly so to
the semi-alpebraical method just described, which seems to have found its first
exponents in Heron and Pappus. Thus the proposition that the differena of
the squares on two straight lines is equal to the rectangle coiUained fy the sum

M-3»4]

PROPOSITIONS 2—4

379

j and the dijfertnce of the straight Unes^ which is generally given as equivalent to

t II. 5, 6, can be proved by means of ii. i, 2, 3, as shown

I by Laixiner. For suppose the given straight lines are ^ 9 P

AB^ BC^ the latter being measured along BA,

Then, by 11. 2, the square on AB is equal to the sum of the rectangles
. AB, BC 9Xi^ AB, AC.

\ By II. 3, the rectangle AB, BC is equal to the sum of the square on BC

i and the rectangle AC, CB,

{ Therefore the square on AB is equal to the square BC together with the

\ sum of the rectangles AC, AB and AC, CB.
\ But, by II. I, the sum of the latter rectangles is equal to the rectande

contained by ^C and the sum of AB, BC, Le. tiie rectangle contained by me

sum and difference of AB, BC.

Hence the square on AB is equal to the square on BC and the rectangle

contained by the sum and difference oi AB, BC:

that is, the difference of the squares on AB, BC is equal to the rectangle

contained by the sum and difference of AB, BC.

\l Proposition 4.

J I/a straight line be cat at random, the square on the whole
' is equal to the squares on the segments and twice the rectangle
contained by the segments.

For let the straight line AB be cut at random at C ;
I say that the square on AB is equal to the squares on AC,
CB and twice the rectangle contained
by AC, CB.

For let the square ADEB be de-
. ^scribed on AB^ [i. 46]

!o let BD be joined ;
* through C let CF be drawn parallel to
either AD or EB,
^ and through G let -^A^ be drawn parallel
to either AB or DE. [i. 31]

Then, since CF is parallel to AD,
and BD has fallen on them,
' the exterior angle CGB is equal to the interior and opposite
angle -^Z?iff. [»• 29]

But the angle ADB is equal to the angle ABD,
since the side BA is also equal to AD ; [i. 5]

therefore the angle CGB is also equal to the angle GBC,
so that the side BC is also equal to the side CG. [i. 6]

A C B

/.

G *

6 i

^ k

38o BOOK II [II. 4

But CB is equal to GK, and CG to KB ; [i. 34]

therefore GK is also equal to KB ;

*5 therefore CGKB is equilateral.

I say next that it is also right-angled.

For, since CG is parallel to BKy

the angles KBC, GCB are equal to two right angles.

[1. 29]
But the angle KBC is right ;

y> therefore the angle BCG is also right,

so that the opposite angles CGK, GKB are also right

[»-34]
Therefore CGKB is right-angled ;

and it was also proved equilateral ;

therefore it is a square ;
35 and it is described on CB.
For the same reason

HF is also a square ;
and it is described on HG^ that is AC. [i. 34]

Therefore the squares HFy A^Care the squares on AC, CB.
40 Now, since AG\% equal to GE,
and AG\^ the rectangle AC, CB, for GC is equal to CB,
therefore GE is also equal to the rectangle AC, CB.

Therefore AG, GE are equal to twice the rectangle AC^
CB.
45 But the squares HF, CK are also the squares on AC, CB;
therefore the four areas NF, CK, AG, GE are equal to
the squares on AC, CB and twice the rectangle contained by *
AC, CB.

But NF, CK, AG, GE are the whole A DEB,
so which is the square on AB.

Therefore the square on AB is equal to the squares on *"
AC, CB and twice the rectangle contained by AC, CB.

Therefore etc. q. e. d. |

s. twice the rectangle contained by the segments. By a carious idiom this is in
Greek '* the rectai^Ie tivice coniainid by the segments." SimiUriy ** twice the rectangle .
contained by ^C CB'* vi expressed as "the rectangle tioiee contained by AC^ CB** {T6 9is
inr6 T&p AT, FB rtpux^/uifaif ^ptfoyiirior).

35i 58* described. 39, 45. the squares (before **on"). These words are not m the
Greek, which simply says that the squares *'are on " {tlalw dw6) their respective sides.

46. areas. It is necessary to supply some substantive (the Greek leaves it to be under-
stood); and I prefer " areas ** to " 6giires."

i

II. 4] PROPOSITION 4 381

The editions of the Greek text which preceded that of E. F. August
(Berlin, 1826 — 9) give a second proof of this proposition introduced by the
usual word aXXa>s or "otherwise thus." Heiberg follows August in omitting
this proof, which is attributed to Theon, and which is indeed not worth
reproducing, since it only differs from the genuine proof in that portion of it
which proves that CGKB is a square. The proof that CGKB is equilateral
is rather longer than Euclid's, and the only interesting point to notice is that,
whereas Euclid still, as in i. 46, seems to regard it as necessary to prove that
all the angles of CGKB are right angles before he concludes that it is right-
angled^ Theon says simply " And it also has the angle CBK right ; therefore
CK is a square." The shorter form indicates a legitimate abbreviation of the
genuine proof; because there can be no need to repeat exactly that part of the
proof of I. 46 which shows that all the angles of the figure there constructed
are right when one is.

There is also in the Greek text a Porism which is undoubtedly interpolated:
"From this it is manifest that in square areas the parallelograms about the
diameter are squares." Heiberg doubted its genuineness when preparing his
edition, and conjectured that it too may have been added by Theon ; but the
matter is placed beyond doubt by a papyrus-fragment referred to already (see
Heiberg, Paralipomena zu Euklid, in Hermes xxxvui., 1903, p. 48) in which
the Porism was evidently wanting. It is the only Porism in Book 11., but
does not correspond to Proclus' remark (p. 304, 2) that "the Porism found fn
the second book belongs to a probletti,'' Heiberg regards these words as
referring to the Porism to iv. 15, the correct reading having probably been not
Scvrcptp but S', i.e. rcrapr<{>.

The semi-algebraical proof of this proposition is very easy, and is of course
old enough, being found in Clavius and in most later editions. It proceeds
thus.

By II. 2, the square on AB is equal to the sum of the rectangles AB^ AC
and AB, CB.

But, by II. 3, the rectangle AB, AC \s equal to the sum of the square on
^Cand the rectangle AC, CB ;
» while, by 11. 3, the rectangle AB, CB is equal to the sum of the square on
BC and the rectangle AC, CB,

Therefore the square on AB is equal to the sum of the squares on
AC, CB and twice the rectangle AC, CB.

The figure of the proposition also helps to visualise, in the orthodox
manner, the proof of the theorem deduced above from 11. i — 3, viz. that the
difference of the squares on two git^en straight lines is equal to the rectangle
contained by the sum and the difference of the lines.

For, if the lines be AB, BC respectively, the shorter of the lines being
measured along BA, the figure shows that

the square A£ is equal to the sum of the square CK and the rectangles

AF,FK\
that is, the square on AB is equal to the sum of the square on BC and

the rectangles AB, ^Cand AC, BC
But the rectangles AB, AC and BC, ACeive, by 11. i, together equal to
the rectangle contained by ^Cand the sum of AB, BC,
i.e. to the rectangle contained by the sum and difference of AB, BC
Whence the result follows as before.

sSa

BOOK II

[n-4.5

The im>position ii. 4 can also be extended to the case where a straight
line is divided into any number of segments ; for the figure will show in like
manner that the square on the whole line is equal to the sum of the squares
on all the parts together with twice the rectangles contained by every pair of
the parts.

Proposition 5.

// a straight line be cut into equal and unequal segments^
the rectangle contained by the unequal segments of the whole
together with the square on the straight line between the
points of section is equal to the square on the haJf.

For let a straight line AB be cut into equal segments
at C and into unequal s^^ents at D ;

I say that the rectangle contained by AD, DB together with
the square on CD is equal to the square on CB.

For let the square CEFB be described on Gff, [i. 46]

and let BE be joined ;

through D let DG be drawn parallel to either CE or BF,
through H again let KM be drawn parallel to either AB or
EF,

and again through A let AK be drawn parallel to either CL
or BM. [i. 31]

Then, since the complement CH is equal to the comple-
ment HF, [i. 43] *^^
let DM be added to each ;

therefore the whole CM is equal to the whole DF.

But CM is equal to AL,

since AC is also equal to CB ; [i. 36] . ^

therefore AL is also equal to DF. <

Let CH be added to each ;

therefore the whole A// is equal to the gnomon NOP.

rr
r

I

II. s] PROPOSITIONS 4, 5 383

But AH is the rectangle AD, DB, for DH is equal to
DB,

therefore the gnomon NOP is also equal to the rectangle
AD, DB.

Let LG, which is equal to the square on CD, be added to
each;

therefore the gnomon NOP and LG are equal to the
rectangle contained by AD, DB and the square on CD.

But the gnomon NOP and LG are the whole square
CEFB, which is described on CB ;

therefore the rectangle contained by AD, DB together
with the square on CD is equal to the square on CB.

Therefore etc. Q. e. d.

3. between the points of section, literally "between the sections^** the word being
the same (ro^^) as that used of a conic section.

It will be observed that the gnomon is indicated in the figure by three separate letters
and a dotted curve. This is no doubt a clearer way of showing what exactly the gnomon is
than the method usual in our text-books. In this particular case the figure of the Mss. has
iwo M*s in it, the gnomon being MNS. I have corrected the lettering to avoid confusion.

It is easily seen that this proposition and the next give exactly the
theorem already alluded to under the last propositions, namely that the
difference of the squares on two straight lines is equal to the rectangle contained
by their sum and difference. The two given lines are, in 11. 5, the lines CB
and CD, and their sum and difference are respectively equal to AD and DB.
To show that n. 6 gives the same theorem we have only to make CD the
greater line and CB the less, i.e. to
draw CD' equal to CB, measure ^ C D B

; CB* along it equal to CD, and then ' '

produce B'C to A', making A'C equal 4J g; f ti

to B'C, whence it is immediately clear
that A'D' on the second line is equal

to AD on the first, while D'B' is also equal to DB, so that the rectangles
- AD, DB and A'D', DF are equal, while the difference of the squares on

CB, CD is equal to the difference of the squares on CD, CB.
'1 Perhaps the most important fact about n. 5, 6 is however their bearing on

the

Geometrical solution of a quadratic equation.

Suppose, in the figure of 11. 5, that AB = a, DB = x\
then tf^ - :r* = the rectangle AH

= the gnomon NOP,

Thus, if the area of the gnomon is given (=^, say), and if a is given
(=AB), the problem of solving the equation

ax^a^=i^
is, in the language of geometry. To a given straight Une (a) to apply a rectangle
which shall be equal to a gitfen square (^) and shall fall short by a square figure,
Le. to construct the rectangle AH ox the gnomon NOP,

Now we are told by Proclus (on i. 44) that '' these propositions are ancient

^

384

BOOK II

[11. 5

D B

/

/

"

//

H

and the discoveries of the Muse of the Pythagoreans, the application of
areas, their exceeding and their falling-short" We can therefore hardly
avoid crediting the Pythagoreans with the geometrical solution, based upon
II. 5, 6, of the problems corresponding to the quadratic equations which
are directly obtainable from them. It is certain that the Pythagoreans solved
the problem in 11. 1 1, which corresponds to the quadratic equation

and Simson has suggested the following easy solution of the equation now in
question,

on exactly similar lines.

Draw CO perpendicular to AB and equal to b\ produce OC to N so
that 0N= CB (or \a)\ and with O as centre
and radius ON describe a circle cutting CB
\nD.

Then DB (or x) is found, and therefore
the required rectangle AH,

For the rectangle AD^ DB together with
the square on CD is equal to the square on
CB, [n. 5]

Le. to the square on OD,
i.e. to the squares on OC, CD] [i. 47]
whence the rectangle AD, DB is equal to the square on OC,
or ojc - :r* = ^.

It is of course a necessary condition of the possibility of a real solution
that ^ must not be greater that {\aY, This condition itself can easily be
obtained from Euclid's proposition ; for, since the sum of the rectangle AD,
DB and the square on CD is equal to the square on CB, which is constant,
it follows that, as CD diminishes, i.e. as D moves nearer to C*, the rectangle
AD, DB increases and, when D actually coincides with C so that CD
vanishes, the rectangle AD, DB becomes the rectangle AC, CB, i.e. the
square on CB, and is a maximum. It will be seen also that the geometrical
solution of the quadratic equation derived from Euclid does not differ from
our practice of solving a quadratic by completing the square on the side
containing the terms in 3^ and x.

But, while in this case there are two geometrically real solutions (because
the circle described with ON as radius will not only cut CB in D but will
also cut AC m another point E), Euclid's figure corresponds to one only of
the two solutions. Not that there is any doubt that Euclid was aware that the
method of solving the quadratic gives two solutions ; he could not fail to see
that X = BE satisfies the equation as well zs x = BD. If however he had
actually given us the solution of the equation, he would probably have
omitted to specify the solution x = BE because the rectangle found by means
of it, which would be a rectangle on the base AE (equal to BD) and with
altitude EB (equal to AD), is really an equal rectangle to that corresponding
to the other solution x = BD ; there is therefore no real object in distinguishing
two solutions. This is easily understood when we regard the equation as a
statement of the problem of finding two magnitudes when their sum (a) and
product (^) are given, i.e. as equivalent to the simultaneous equations

x-^y^a,
xy^V.

I

II. 5, 6] PROPOSITIONS 5, 6 385

These symmetrical equations have really only one solution, as the two apparent
solutions are simply the result of interchanging the values of x and y. This
form of the problem was known to Euclid, as appears from the J^afa^ Prop.
85, which states that, If two straight lines contain a parallelogram given in
magnitude in a given angle^ and if the sum of them be given, then shall each
of them be given.

This proposition then enables us to solve the problem of finding a
rectangle the area and perimeter of which are both given ; and it also enables
us to infer that, of all rectangles of given perimeter, the square has the
greatest area, while, the more unequal the sides are, the less is the area.

If in the figure of 11. 5 we suppose that AD- a, BD = b, we find that
CB = {a'^b)l2 and CZ> = (a-^)/2, and we may state the result of the
proposition in the following algebraical form

(i±i)-.(«^y.^.

This way of stating it (which could hardly have esca()ed the Pythagoreans)
gives a ready means of obtaining the two rules, respectively attributed to the
Pythagoreans and Plato, for finding integral square numbers which are the
sum of two other integral square numbers. We have only to make ab a
perfect square in the above formula. The simplest way in which this can be
done is to put a^n\ b=^i, whence we have

1 (=^)'-(=^y-<

and in order that the first two squares may be integral n^, and therefore n,
must be odd. Hence the Pythagorean rule.
Suppose next that a = 2f^y ^ = 2, and we have
(«*+i)«-(««-i)» = 4««.
whence Plato's rule starting from an even number 2n,

Proposition 6.

// a straight line be bisected and a straight line be added
to it in a straight line, the rectangle contained by the whole
with the added straight line and the added straight line together
with the square on the half is equal to the square on the
straight line' made up of the half and the added straight
line.

For let a straight line AB be bisected at the point C, and
let a straight line BD be added to it in a straight line ;
. I say that the rectangle contained by AD, DB together

[ with the square on CB is equal to the square on CD.

For let the square CEFD be described on CD, [i. 46]

and let DE be joined ;

through the point B let BG be drawn parallel to either EC or
DF,

H. K. »5

386

BOOK II

[11.6

through the point H let KM be drawn parallel to either AB
or EF,

and further through A let AK
be drawn parallel to either CL
or DM. [i. 31]

Then, since AC is equal
toC^,

AL is also equal to C//. [i. 36]
But CN is equail to //F. [i. 43]
Therefore AL is also equal
to//F.

Let Gl/ be added to each ;

therefore the whole AM is equal to the gnomon NOP.
But AM is the rectangle AD, DB,

for DM is equal to VB ;

therefore the gnomon NOP is also equal to the rectangle
AD, DB.

Let LG^ which is equal to the square on BC^ be added
to each ;

therefore the rectangle contained by AD, DB together
with the square on CB is equal to the gnomon NOP and LG.

But the gnomon NOP and LG are the whole square
CEFD, which is described on CD ;

therefore the rectangle contained by AD, DB together
with the square on CB is equal to the square on CD.

Therefore etc.

Q. E. D.

In this case the rectangle AD, DB is "a rectangle applied to a given
straight line {AS) but exatding by a square (the side of which is equal to
BD) " ; and the problem suggested by 11. 6 is to find a rectangle of this
description equal to a given area, which we will, for convenience, suppose to
be a square ; Le., in the language of geometry, /o apply to a given straight
line a rectangle which shall he equal to a gwen square and shall exceed hy a
square figure.

We suppose that in Euclid's figure AB^a, BD-x\ then, if the given
square be Ir, the problem is to solve geometrically the equation

The solution of a problem theoretically equivalent to the solution of a
quadratic equation of this kind is presupposed in the fragment of Hippocrates'
Quadrature of lunes preserved in a quotation by Simplidus (Comment, in

II. 6]

PROPOSITION 6

387

I

Aristot, Phys. pp. 6i — 68, ed. Diels) from Eudemus' History of Geometry. In
this fragment Hippocrates (5th cent b.c) assumes the following construction.

AB being the diameter and O the centre of a semicircle, and C being the
middle point of OB and CD at right
angles to AB^ a straight line of length
such that its square is i| times the square
on the radius (i.e. of length a^\, where
a is the radius) is to be so placed, as EFy
between CD and the circumference AD
that it "verges towards B^' that is, EF
when produced passes through B.

Now the right-angled triangles BFC^
BAE are similar, so that

BF\BC^BAxBE,

and therefore the rectangle BE, BF= rect BA, BC

= sq. on BO.

In other words, EF ( = a J\) being given in length, BF ( = *, say) has
to be found such that

(V|a + «):r = a*;
or the quadratic equation

has to be solved.

A straight line of length a^\ would easily be constructed, for, in the
figure, CIJ^ = AC.CB^\c^, or CD=\aJi, and aj% is the diagonal of
a square of which CD is the side.

There is no doubt that Hippocrates could have solved the equation by
the geometrical construction given below, but he may have contemplated, on
this occasion, the merely nuchaniccU process of placing the straight fine of the
length required between CD and the circumference AD and moving it until
E, F, B were in a straight line. Zeuthen {Die Lehre von den Kegelschnitten
im Altertum, pp. 270, 271) thinks tliis probable because, curiously enough,
the fragment speaks immediately afterwards of "joining B to F'^

To solve the equation

we have to find the rectangle AH, or the
gnomon NOP, which is equal in area to A* and
has one of the sides containing the inner right
angle equal to CB or \a. Thus we know
{\af and ^, and we have to find, by i. 47,
a square equal to the sum of two given
squares.

To do this Simson draws BQ at right
angles to AB and equal to b, joins CQ and,
with centre C and radius CQ, describes a
circle cutting AB produced in D, Thus
BD, or X, is found.

Now Uie rectangle AD, DB together with the square on CB

is equal to the square on CD,

Le. to the square on CQ,

Le. to the squares on CB, BQ.

2S—2

388 BOOK II [IL 6, 7

Therefore the rectangle ADi DB is equal to the square on BQ^ that is,

From Euclid's point of view there would only be one solution in this case.
This proposition enables us also to solve the equation

in a similar manner.

We have only to suppose that AB =■ a, and AD (instead of BD) = x \ then
^ ^ ojc = the gnomon.
To find the gnomon we have its area (A*) and the area, CB^ or (Ja)*, by
which the gnomon differs from C7^. Thus we can find D (and therefore
AD or x) by the same construction as that just given.

Converse propositions to 11. 5, 6 are given by Pappus (vii. pp. 948 — 950)
among his lemmas to the Conies of Apollonius to the effect that,
(i) if /> be a point dividing AB unequally, and C another point on AB
such that the rectangle AD^ DB together with the square on CD is
equal to the square on AC^ then

^Cis equal to CB\

(2) if Z? be a point on AB produced, and C a point on AB such that the
rectangle AD^ DB together with the square on CB is equal to the
square on CD^ then

AC is equal to CB.

Proposition 7.

If a straight line be cut at random^ the square an the
whole and that an ane of the segments both together are equal
to twice the rectangle contained by the whole and the said
segment and the square an the remaining segment.

For let a straight line AB be cut at random at the point C\

I say that the squares on AB^ BC are equal to twice the
rectangle contained by AB, BC and the
square on CA.

For let the square ADEB be
described on AB, [146]

and let the figure be drawn.

Then, since -^G^ is equal to GE, [i. 43]
let CF be added to each ;

therefore the whole AF is equal to
the whole CE.

Therefore AF, CE are double of
AF.

But AF, CE are the gnomon KLM znd the square CF\
therefore the gnomon KLM and the square CF are double
olAF.

T

I

i

II 7. 8] PROPOSITIONS 6—8 389

But twice the rectangle AB, BC is also double of AF\
for BF IS equal to BC ;

therefore the gnomon KLM and the square CF are equal to
twice the rectangle AB, BC.

Let DG, which is the square on AC, be added to each ;
therefore the gnomon KLM and the squares BG, GD are
equal to twice the rectangle contained by AB, BC and the
square on AC

But the gnomon KLM and the squares BGy GD are the
whole A DEB and CF,

which are squares described on AB, BC ;
therefore the squares on AB, BC are equal to twice the
rectangle contained by AB^ BC together with the square on
AC.

Therefore etc.

Q. E. D.

An interesting variation of the form of this proposition may be obtained by
regarding AB, BC as two given straight lines of which AB is the greater, and
^C as the difference between the two straight lines. Thus the proposition
shows that the squares on two straight lines are together equal to twice the
rectangle contained by them and the square on their difference. That is, the
square on the difference of two straight lines is equal to the sum of the squares on
the straight lines diminished by twice the rectangle contained by them. In other
words, just as 11. 4 is the geometrical equivalent of the identity

(tf + ^)*=a* + ^ + tab,
so II. 7 proves that

The addition and subtraction of these formulae give the algebraical equivalent
of the propositions 11. 9, 10 and 11. 8 respectively ; and we have accordingly
a suggestion of alternative methods of proving those propositions.

Proposition 8.

If a straight line be cut at random, four times the rectangle
contained by the whole and one of the segments together with
the square on the remaining segment is equal to the square
described on the whole and the aforesaid segment as on one
straight line.

For let a straight line AB be cut at random at the point C;

I say that four times the rectangle contained by AB, BC
together with the square on ^C is equal to the square
described on AB, BC as on one straight line.

39®

BOOK II

[11.8

M

..-

:t/

Q

\

/

K \ '

8 y

^

ft / '

/^

-.

I H 1

L F

For let [the straight line] BD be produced in a straight
line [with AB\ and let iff/? be ^
made equal to CB ;
let the sqmre A £FD be described
on -r4Z?, and let the figure be
drawn double.

Then, since CB is equal to BD,
while CB is equal to GJC, and
BD to ^A^,
therefore G^A' is also equal to KN.

For the same reason
QR is also equal to jRP.

And, since BC is equal to BD, and GKxo KN,
therefore CK is also equal to KD, and G^jff to RN. [i. 36]

But C/T is equal to RN, for they are complements of the
parallelogram CP ; [i. 43]

therefore KD is also equal to GR ;

therefore the four areas DK, CK, GR, RN are equal to one
another.

Therefore the four are quadruple of CK.

Again, since CB is equal to BD,
while BD is equal to BK, that is CG,
and CB is equal to GK, that is GQ,

therefore CG is also equal to GQ.

And, since CG is equal to GQ, and QR to RP,

AG is also equal to MQ, and j2-^ to jff/; [i. 36]

But MQ is equal to QL, for they are complements of the
parallelogram ML ; [i. 43]

therefore -^G^ is also equal to RF;
therefore the four areas AG, MQ, QL, RF are equal to one
another.

Therefore the four are quadruple oi AG.
But the four areas CK, KD, GR, RN were proved to be
quadruple of CK\

therefore the eight areas, which contain the gnomon
STU, are quadruple of AK.

Now, since AK is the rectangle AB, BD, for BK is equal
to BD,

\

II. 8]

PROPOSITION 8

391

therefore four times the rectangle AB, BD is quadruple of
AK.

But the gnomon STC/v/sls also proved to be quadruple

therefore four times the rectangle AB, BD is equal to the
gnomon STC/.

Let O//, which is equal to the square on AC, he added
to each ;

therefore four times the rectangle AB, BD together with
the square on ACis equal to the gnomon STC/ SLtid OH.

But the gnomon STU and OH are the whole square
AEFD,

which is described on AD ;
therefore four times the rectangle AB, BD together with
the square on -^C is equal to the square on AD.

But BD is equal to BC ;
therefore four times the rectangle contained by AB^ BC
together with the square on -^C is equal to the square on
AD^ that is to the square described on AB and BC as on
one straight line.

Therefore etc, ^ ,, ^

Q. E. D.

B

M

This proposition is quoted by Pappus (p. 428, ed. Hultsch) and is used
also by Euclid himself in the Data^ Prop. 86. Further, it is of decided use
in proving the fundamental property of a parabola.

Two alternative proofs are worth giving.

The first is that suggested by the consideration mentioned in the last
note, though the proof is old enough, being given by Clavius and others. It
is of the semi-algebraical type.

Produce AB to D (in the figure of the pro-
position), so that BD is equal to BC,

By II. 4, the square on AD is equal to the
squares on A By BD and twice the rectangle AB,
BD, i.e. to the squares on AB, BC and twice
the rectangle AB, BC.

By II. 7, the squares on AB, BC 9J^ equal to
twice the rectangle AB, BC together with the
square on AC

Therefore the square on AD is equal to four
times the rectangle AB, BC together with the
square on AC

The second proof is after the manner of Euclid but with a difference.
Produce BA to 2> so that AD is equal to BC On BD construct the square
BEFD.

tr

392 BOOK II [ii. 8, 9

Take BG^ EH, FK each equal to BC or AD, and draw ALP, HNM
parallel to BE and GML, ^/W parallel to BD.

Then it can be shown that each of the rectangles BL^ AK, FN, EM is
equal to the rectangle AB, BC, and that PM\% equal to the square on AC,

Therefore the square on BD is equal to four times the rectangle AB,
BC together with the square on AC

Proposition 9.

If a straight line be cut into equal and unequal segments,
the squares on the unequal segments of the whole are double
of the square on the half and of the square on the straight line
between the points of section.

For let a straight line AB be cut into equal segments
at C, and into unequal segments at D\

I say that the squares on AD, DB are double of the
squares on AC, CD.

For let CE be drawn from
C at right angles to AB,
and let it be made equal to
either -^Cor CB \
let EA, EB be joined,
let DF be drawn through D
parallel to EC,

and FG through F parallel to
AB,
and let AF be joined.

Then, since -^C is equal to CE,
the angle EAC is also equal to the angle A EC.

And, since the angle at C is right,

the remaining angles EAC, A EC are equal to one
right angle. [i. 32]

And they are equal ;

therefore each of the angles CEA, CAE is half a right
angle.

For the same reason

each of the angles CEB, EBC is also half a right angle ;

therefore the whole angle AEB is right

And, since the angle GEF is half a right angle.

11. 9] PROPOSITIONS 8, 9 393

and the angle EGF is right, for it is equal to the interior and
1 1 opposite angle ECBy [i. 29]

\ ^ the remaining angle EFG is half a right angle ; [i. 32]

I therefore the angle GEF is equal to the angle EFG^

I so that the side EG is also equal to GF. [i. 6]

Again, since the angle at B is half a right angle,
and the angle FDB is right, for it is again equal to the interior
and opposite angle ECB^ [i. 29]

the remaining angle BFD is half a right angle ; [i. 32]
therefore the angle at B is equal to the angle DFB,

so that the side FD is also equal to the side DB. [i. 6]
Now, since -^C is equal to CE,
the square on -^C is also equal to the square on CE ;
therefore the squares on AC, CE are double of the square
on AC.

But the square on EA is equal to the squares on AC^ CE,
for the angle ACE is right ; [i. 47]

therefore the square on EA is double of the square on AC
Again, since EG is equal to GF,
the square on EG is also equal to the square on GF\

therefore the squares on EG^ GF are double of the square on
GF.

But the square on EF is equal to the squares on EG, GF\
\ therefore the square on EF is double of the square on GF.
\ But GF is equal to CD ; - [i. 34]

^ therefore the square on EF is double of the square on CD.
U But the square on EA is also double of the square on AC\

I therefore the squares on AE, EFzxt, double of the squares

on AC, CD.

And the square on AF is equal to the squares on AE, EF,
for the angle AEF is right ; [i. 47]

therefore the square on AF is double of the squares on AC,
CD.

But the squares on AD, DF are equal to the square on
AF, for the angle at D is right ; [i. 47]

therefore the squares on AD, DF are double of the squares
on AC, CD.

394

BOOK U

[tug

And DFis equal to DB;
therefore the squares on AD, DB are double of the squares
on AC^ CD.

Therefore etc.

Q. E. D.

It is noteworthy that, while the first eight propositions of Book ii. are
proved independently of the Pythagorean theorem i. 47, all the remaining
propositions beginning with the 9th are proved by means of it Also the 9th
and loth propositions mark a new departure in another respect ; the method
of demonstration by showing in the figures the various rectangles and squares
to which the theorems relate is here abandoned.

llie 9th and loth propositions are related to one another in the same way
as the 5th and 6th ; they really prove the same result which can, as in the
earlier case, be comprised in a single enunciation thus : The sum of the squares
on the sum and difference of tufo given straight lines is equal to twice the sum of
the squares on the lines.

The semi-algebraical proof of Prop. 9 is that suggested by the remark on
the algebraical formulae given at the end of the note on 11. 7. It implies
with a very slight modification to both 11. 9 and 11. 10. We wiU put in
brackets the variations belonging to 11. 10.

The first of the annexed lines is the figure ^ O D B

for II. 9 and the second for 11. 10. '

By II. 4, the square on AD is equal to a B O

the squares on AC, CD and twice the » »

rectangle AC^ CD.

By II. 7, the squares on CB, CD {CD, CB) are equal to

twice the rectangle CB, CD together with the square on BD.

By addition of these equals crosswise,
the squares on AD, DB together with twice the rectangle CB, CD are
equal to the squares on AC, CD, CB, CD together with twice
the rectangle AC, CD.

But AC, CB are equal, and therefore the rectangles AC, CD and CB,
CD are equal.

Taking away the equals, we see that

the squares on AD, DB are equal to the squares on AC, CD, CB, CD,

Le. to twice the squares on AC, CD.
To show also that the method of geometrical algebra illustrated by
II. I — 8 is still effective for the purpose of
proving 11. 9, 10, we wiU now prove 11. 9 in
that manner.

Draw squares on AD, DB respectively
as shown in the figure. Measure DH along
DE equal to CD, and HL along HE also
equal to CD.

Draw HK, LNO parallel to EF, and
CNM parallel to DE.

Measure NP along NO equal to CD,
and draw FQ parallel to DE.

r

1

'1

Q M E

II. 9i xo]

PROPOSITIONS 9, lo

39S

Now, since AD^ CD are respectively equal to DE^ DH^
HE is equal to j4C or CB;
and, since HZ is equal to CZ>, L£ is equal to DB.

Similarly, since each of the segments EAf, MQ is equal to CD,
EQ is equal to EZ or BD.

Therefore OQ is equal to the square on DB.

We have to prove that the squares on AD, DB are equal to twice the
squares on AC, CD.

Now the square on AD includes XM (the square on AC) and CH, HN
(that is, twice the square on CD\

Therefore we have to prove that what is left over of the square on AD
together with the square on DB is equal to the square on AC.

The parts left over are the rectangles CK and NE, which are equal to
KN, PM respectively.

But the latter with the square on DB are equal to the rectangles KN,
/Wand the square OQ,

i.e. to the square KM, or the square on AC.

Hence the required result follows.

Proposition id.

If a straight line be bisected, and a straight line be added
to it in a straight line^ the square on the whole with the added
straight line and the square on the added straight line both
together are double of the square on the half and of the square
described on the straight line made up of the hcUf and the
added straight line as on one straight line.

For let a straight line AB be bisected at C, and let a
straight line BD be added to it in a straight line ;

I say that the squares on AD, DB are double of the
squares on AC, CD.

For let CE be drawn from
the point C at right angles to
AB [i. ii], and let it be made
equal to either -^C or CB [i. 3] ;

let EA, EB be joined ;

through E let EF be drawn
parallel to AD,

and through D let FD be drawn
parallel to CE. [i. 31]

Then, since a straight line EF falls on the parallel straight
lines EC, FD,

396 BOOK II [ii. lo

the angles CEF, EFD are equal to two right angles; [i. 89]
therefore the angles FEB, EFD are less than two right
angles.

But straight lines produced from angles less than two
right angles meet ; [i. Post. 5]

therefore EB, FD, if produced in the direction B, D, will
meet.

Let them be produced and meet at G,
and let -^G^ be joined.

Then, since -^C is equal to CE,
the angle EAC is also equal to the angle A EC ; [i. 5]

and the angle at C is right ;

therefore each of the angles EAC, A EC is half a right
angle. [i. 3a]

For the same reason

each of the angles CEB, EBC is also half a right angle ;
therefore the angle AEB is right

And, since the angle EBC is half a right angle,
the angle DBG is also half a right angle. [1. 15]

But the angle BDG is also right,
for it is equal to the angle DCE, they being alternate; [i. 89]

therefore the remaining angle DGB is half a right angle ;

[I. 3a]
therefore the angle DGB is equal to the angle DBG,

so that the side BD is also equal to the side GD. [i. 6]

Again, since the angle EGF is half a right angle,

and the angle at F is right, for it is equal to the opposite

angle, the angle at C, [i. 34]

the remaining angle FEG is half a right angle ; [i. 3a]

therefore the angle EGF is equal to the angle FEG,

so that the side GF is also equal to the side EF. [i. 6]
Now, since the square on EC is equal to the square on
CA,

the squares on EC, CA are double of the square on CA.
But the square on EA is equal to the squares on EC, CA ;

.['.47]
therefore the square on EA is double of the square on AC.

\C. N. i]

1

II. lo] PROPOSITION lo 397

Again, since FG is equal to EF^
the square on FG is also equal to the square on FE ;
therefore the squares on GF, FE are double of the square on
EF.

But the square on EG is equal to the squares on GF^ FE\

[1.47]

therefore the square on EG is double of the square on EF.
And EF is equal to CD ; [i. 34]

therefore the square on EG is double of the square on CD.

But the square on EA was also proved double of the square

on AC\

therefore the squares on AEy EG are double of the squares
on AC, CD.

And the square on -^G^ is equal to the squares on AE,
EG, [1.47]

therefore the square on AGxs double of the squares on AC,
CD.

But the squares on AD, DG are equal to the square on -^G^ ;

['• 47]
therefore the squares on AD, DG are double of the squares
on AC^ CD.

And DG is equal to DB ;

therefore the squares on AD^ DB are double of the squares
on AC, CD.

Therefore etc.

Q. E. D.

The alternative proof of this proposition by means of the principles
exhibited in 11. 1—8 follows the lines of that
which I have given for the preceding proposition. ^ q b D

It is at once obvious from the figure that the
square on AD includes within it twice the square
on ^C together with once the square on CD.
What is left over is the sum of the rectangles AH,
KE. These, which are equivalent to BH, GK,
make up the square on CD less the square on
BD. Adding therefore the square BG to each
side, we have the required result.

Another alternative proof of the theorem which
includes, both 11. 9 and 10 is worth giving. The
theorem states that the sum of the squares on the

sum and difference of two given straight lines is equal to twice the sum of the
squares on the lines.

H L_i

K

Q H
L K

39S BOOK II [11. 10

Let AD^ DB be the two given straight lines (of which AD is die greater),
placed so as to be in one straight line. Make AC equal to DB and com-
plete the figure as shown, each of the segments CG
and DH bsing equal to AC or DB. A O D B

Now, AD^ DB being the given straight lines, AB
is their sum and CD is equal to their difference.

Also AD is equal to BC

And AE is the square on AB^ GK is equal to
the square on CD^ AK or FH\& the square on AD^
and BL the square on CB^ while each of the small
squares AG^ BH^ EK^ FL is equal to the square on
ACoiDB.

We have to prove that twice the squares on AD^
DB are equal to the squares on AB^ CD,

Now twice the square on AD is the sum of the squares on AD^ CB^
which is equal to the sum of the squares BJL^ FH\ and the figure shows
these to be equal to twice the inner square GK and once the remainder of
the large square AE excluding the two squares AG^ KE^ which latter squares
are equal to twice the square on ^C or DB,

Therefore twice the squares on AD^ DB are equal to twice the inner
square GK together with once the remainder of the laige square AE^ that is,
to the sum of the squares AE^ GK^ which are the squares on AB^ CD,

** Side" and '' diagonal" numbers giving successive approxi-
mations to ^2.

2^uthen pointed out (Die Lehre van den Kegeischnitien m Altertum^ 1886,
pp. 27, a8) that 11. 9, 10 have great interest

m connexion with a problem of indeterminate r g g ^

analysis which received much attention from

the ancient Greeks. If we take the straight line AB divided at C and D as

in II. 9, and if we put CD=x^ DB=y^ the result obtained by Euclid, namely :

^27» + Z>i5» = 2^C« + 2CZ>»,

or AL^ - 2AO ^ 2CL^ - DB^,

becomes the formula

(2x -^yf - 2 (jf +>)■ = 2s^ -y.

If therefore x,> be numbers which satisfy one of the two equations

the formula gives us two higher numbers, x -k-y and 2x +>, which satisfy the
other of the two equations.

Euclid's propositions thus give a general proof of the very formula used
for the formation of the succession of what were caUed "xiir/^" and ^^diagpnai
numbers."

As is well known, Theon of Smyrna (pp. 43, 44, ed. Hiller) describes this
system of numbers. The unit, being the beginning of all things, must be
potentially both a side and a diameter. Consequently we begin widi two units,
the one being the first side and the other the first diameter^ and (a) firom the
sum of them, (b) from the sum of twice the first unit and once the second, we
form two new numbers

1.1 + 1 = 2, 2.1 + 1 = 3.

II. lo] PROPOSITION lo 399

Of these new numbers the first is a side- and the second a ^/Vi^^oZ-number,
or (as we may say)

In the same way as these numbers were formed from a^ = i, ^j = i, successive
pairs of numbers are formed from Oa, d^^ and so on, according to the formula

Thus V «f=2 + 3 = 5, ^,= 2.2 + 3 = 7,

^4 = 5 + 7 = ", ^4=2.5 + 7 = 17,
and so on.

Theon states, with reference to these numbers, the general proposition that

and he observes (i) that the signs alternate as successive ^s and 0*3 are taken,
d^ - 2tfi' being equal to - i, <^* - 2a} equal to + i, ^,' - 2a} equal to - i, and
so on, (2) that the sum of the squares of all the ^'s will be double of the sum
of the squares of all the a's. [If the number of successive terms in each
series is finitty it is of course necessary that the number should be even.]
The proof, no doubt omitted because it was well known, may be put
algebraically thus

d^f - 2a^ = (2a^^ + ^,_i)» - 2 (a»-, + ^,.,)*
= 2a..,»-^«.,«
= -K-i'-2a,.,»)
= + (^»-a* — 2a,-a'), in like manner,
and so on, while d^ - 2a f = - i. Thus the theorem is established.

Euclid's propositions enable us to establish the theorem geometrically;
and this fact might well be thought to confirm the conjecture that the
investigation of the indeterminate equation 20^ - J'' = ± i in the manner
explained by Theon was no new thing but began at a period long before
Euclid's time. No one familiar with the truth of the proposition stated by
Theon could have failed to observe that, as the corresponding side- and
^la^ifoAnumbers were successively formed, the value of d^\a^ would
approach more and more nearly to 2, and consequently that the successive
fractions d^a^ would give nearer and nearer approximations to the value of
V2, viz. i, |, I, U, ^,....

It IS fiurly clear that m the famous passage of Plato's Republic ^546 c)
about the "geometrical number" some such system of approximations is
hinted at Plato there contrasts the *^raHonal diameter of five (^i^^ Sca/Acrpof
n7S xcfixa&K) with the ** irrational " (diameter). This was certainly taken
from the Pythagorean theory of numbers (cf. the expression immediately
preceding, 546 B, C irirra wpwrr/yopa. #cai ^i^^ irpo« oAXi^Xa airc^i^vav, with the
phrase iravra yvwrra #cai wordyopa aXXaXoi^ dirtpyaitrai. in the figment of
Philolaus). The reference of Plato is to the following consideration. If the
square of side 5 be taken, the diagonal is V2. 25 or ijjo. This is the
Pythagorean " irraii onal d iameter " of 5; and the "rational diameter" was
the i^proximation n/ 50 - i, or 7.

But the conjecture of Zeuthen, and the attribution of the whole theory of
side- and ^^.^omAnumbers to the Pythagoreans, have now been fully confirmed
by the publication of KroU's edition of Procli Diadochi in Platonis rempublicam
iommentarii {^tx\msx\ Vol 11., 1901. The passages ([cc. 23 and 27, pp. 24,
95 and 27 — 29) which there saw the light for the first time describe the same

400

BOOK II

[n. lo

83rstein of forming side- and diagtmal'r\\ya!\xx^ and definitely attribute it, as
well as the distinction between the ''rational" and "irratioiud diameter," to
the Pythagoreans. Proclus further says (p. 27, 16 — 22) that the property of the
side- and dtagonal-iixxxsAxx^ *^ is proved graphically (ypafifumSf ) in the second
book of the Elements by ^him' (ax* Ijccivov). For, if a straight line be hisuted
and a straight line be added to it^ the square on the whole line including the
added straight line and the square on the latter alone are double of the square on
the half of the original straight line and of the square on the straight Une made
up of the half and of the added straight line^ And this is simply Eud. 11. 10.
Proclus then goes on to show specifically how this proposition was used to
prove that, with the notation above used, the diameter corresponding to the
side a -^ d is za + d. Let AB be a side and BC equal to it, while CD is the
diameter corresponding to AB^ i.e. a straight line such that the square on it is
double of the square on AB. (I use the figure supplied by Hultsch on p. 397
of Kroll's VoL II.)

Then, by the theorem of EucL 11. 10, the squares on AD^ DCtxt double
of the squares on AB^ BD.

But the square on DC (i.e. BE^ is double of the square on AB\ therefore,
by subtraction, the square on AD is double of the square on BD.

And the square on DF^ the diagonal corresponding to the side BD^ is
double the square of BD.

Therefore the square on DF is equal to the square on AD^ so that DF\&
equal to AD.

That is, while the side BD is, with our notation, a -»• i^ the corresponding
diagonal^ being equal to AD, is 2a -»• d.

In the above reference by Proclus to 11. 10 dx' Uti^ov "by him^ must
apparently mean vx* EvicXciSov, '' by Euclid," although Euclid's name has not
been mentioned in the chapter; the phrase would be equivalent to saying
"in the second Book of the fomous Elements." But, when Proclus sajrs "this
is proved in the second Book of the Elements," he does not imply that it had
not been proved before ; on the contrary, it is clear that the theorem had
been proved by the Pythagoreans, and we have therefore here a confirmation
of the inference from the part played by the gnomon and by i. 47 in Book 11.
that the whole of the substance of that Book was Pythagorean. For further
detailed explanation of the passages of Proclus reference should be made to
Hultsdi's note in Kroll's VoL 11. pp. 393—400, and to the separate article,
also by Hultsch, in the Bibliotheca iiathematica i„ 1900, pp. 8—12.

P. Ba^h has an ingenious suggestion (see Zeitschr^fur Math. m. Phytih

11. lo]

PROPOSITION xo

401

XXXI. Hist-litt Abt. p. 135, and Cantor, Geschichte der Maihenuitik^ i„ p. 437)

as to the way in which the formation of the successive

side- and ^iVi^/ia/-numbers may have been discovered,

namely by observation from a very simple geometrical

figure. Let ABC be an isosceles triangle, right-angled $it

A^ with sides 0,^.1, a..!, ^_i respectively. If now the

two sides AB^ AC about the right angle be lengthened

by adding ^.-i to each, and the extremities D^ E ht

joined, it is easily seen by means of the figure (in which

BF^ CG are perpendicular to DE) that the new diagonal

d^ is equal to 2a«_i + ^_i, while the equal sides a^ are, by construction, equal

to a,-i + ^,.1.

Important deductions from II. 9, zo.

I. Pappus (vii. pp. 856 — 8) uses 11. 9, 10 for proving the well-known
theorem that

Tht sum of the squares on two sides of a triangle is equal to twice the square
on half the base together with twice the square on the straight line joining the
middle point of the base to the opposite vertex.

Let ABC be the given triangle and D the middle point of the base BC
Join AD^ and draw AE perpendicular to BC (produced if necessary).

C E

Now, by II. 9, 10,
the squares on BE^ EC are equal to twice the squares on BD^ DE.
Add to each twice the square on AE.
Then, remembering that

the squares on BE^ EA are equal to the square on BA^
the squares on AE^ EC are equal to the square on A C,
and the squares on AEy ED are equal to the square on AD^
we find that

the squares on BA^ AC art equal to twice the squares on AD, BD.
The proposition is generally proved by means of 11. 12, 13, but not, I
think, so conveniently as by the method of Pappus.

II. The inference was early made by Gregory of St Vincent (1584-1667)
and Viviani (162 2-1 703) that /n any parallelogram the squareron the diagonals
are together equal to the squares on the sides, or to twice the squares on adjacent
sides.

III. It appears that Leonhard Euler (1707-83) was the first to discover
the corresponding theorem with reference to any quadrilateral, namely that
In any quadrilateral the sum of the squares on the sides is equal to the sum of the
squares on the diagonals and four times the square on the line joining the middle

H. K.

26

403

BOOK II

[ll. lO, It

faints of the diagonals. Euler seems however to have proved the property
from the corresponding theorem for paralldograros just quoted (cf. CamerePs
Euclid, Vol I. pp. 468, 469) and not from the property of the triangle, though
the latter brings out the result more easily.

H B

(5 K D

Proposition ii.

To cut a given straight line so that the rectangle contained
by the whole and one of the segments is equal to the square on
the remaining segment.

Let AB be the given straight line ;

thus it is required to cut AB so that the rectangle contained
by the whole and one of the segments is
equal to the square on the remaining
segment

For let the square ABDC be described
on AB ; [i. 46]

let y^C be bisected at the point E, and let
BE be joined ;

let CA be drawn through to /% and let EF
be made equal to BE ;

let the square FH be described on AF, and
let GH be drawn through to K.

I say that AB has been cut at H so as to make the
rectangle contained by AB^ BH equal to the square on AH.

For, since the straight line AC has been bisected at E,
and FA is added to it,

the rectangle contained by CF, FA together with the
square on AE is equal to the square on EF. [n. 6]

But EF is equal to EB ;

therefore the rectangle CF, FA together with the square
on AE is equal to the square on EB.

But the squares on BA, AE are equal to the square on
EB, for the angle at A is right ; [l 47]

therefore the rectangle CF, FA together with the square
on AE is equal to the squares on BA, AE.

Let the square on AE be subtracted from each ;
therefore the rectangle CF, FA which remains is equal to
the square on AB.

I

M

r II. II, 1 a] PROPOSITIONS lo— la 403

! 1 Now the rectangle CF, FA is FK, for AF is equal to

[1 FG',

and the square on AB is AD ;
; • therefore /^K is equal to AD.

Let AJC be subtracted from each ;
therefore /**// which remains is equal to ND.
And I/D is the rectangle AB, BH, for y^-ff is equal to
BD\
and /T^ is the square on AH\

therefore the rectangle contained by AB, BH is equal
to the square on HA.

therefore the given straight line AB has been cut at H
so as to make the rectangle contained by AB, BH equal to
the square on HA.

Q, E. F.

As the solution of this problem is necessary to that of inscribing a regular
pentagon in a circle (Eucl. rv. lo, ii), we must necessarily conclude that it
was solved by the Pythagoreans, or, in other words, that they discovered the
geometrical solution of the quadratic equation

a (a - jc) = jc*,

or *** + A3C = fl*.

The solution in ii. ix, too, exactly corresponds to the solution of the more
general equation

which, as shown above (pp. 387 — 8), Simson based upon n. 6. Only Simson's
I solution, if applied here, gives us the point Fon CA produced and does not
directly find the point H. It takes E the middle point of CA, draws AB at
right angles to CA and of length equal to CA, and then describes a circle
with EB as radius cutting EA produced in F. The only difference between
the solution in this case and in the more general case is that AB is here equal
to CA instead of being equal to another given straight line b.

As in the more general case, there is, from Eudid's point of view, only one
solution.

The construction shows that CF is also divided at ^ in the manner
described in the enunciation, since the rectangle CF^ FA is equal to the
square on CA,

The problem in 11. 11 reappears in vi. 30 in the form of cutting a given
straight Hne in extreme and mean ratio.

Proposition 12.

. In obtus^-an^led triangles the square on the side subtending
the obtuse angle is greater than the squares on the sides con-
taining the obtuse angle by twice the rectangle contained by one
of the sides about the obtuse angle, namely that on which the

26—3

404 BOOK II [ii. 12

perpendicular falls, and the straight line cut off outside by the
perpendicular towards the obtuse angle.

Let ABC be an obtuse-angled triangle having the angle
BAC obtuse, and let BD be drawn from the point B per-
pendicular to CA produced ;

I say that the square on BC is greater than the squares
on BA, AC hy twice the rectangle con-
tained by CA, AD.

For, since the straight line CD has
been cut at random at the point A,
the square on DC is equal to the
squares on CA, AD and twice the rect-
angle contained by CA, AD. [n. 4]

Let the square on DB be added to
each ;

therefore the squares on CD, DB are equal to the squares on
CA, AD, DB and twice the rectangle CA, AD.

But the square on CB is equal to the squares on CD, DB,
for the angle at D is right ; [i. 47]

and the square on AB is equal to the squares on AD,
DB, [L47]

therefore the square on CB is equal to the squares on CA, AB
and twice the rectangle contained by CA, AD ;

so that the square on CB is greater than the squares on
CA, AB by twice the rectangle contained by CA, AD.

Therefore etc. q. e. d.

Since in this proposition and the next we have to do with the squares on
the sides of triangles, the particular form of graphic representation of areas
which we have had in Book n. up to this point does not help us to visualise
the results of the propositions in the same way, and only two lines of proof
are possible, (i) by means of the results of certain earlier propositions in
Book n. combined with the result of i. 47 and (2) by means of the procedure
in Euclid's proof of i. 47 itself. The alternative proofs of n. la, 13 after the
manner of Euclid's proof of i. 47 are therefore alone worth giving.

These proofs appear in certain modem text-books (e.g. Mehler, Henrid and
Treutlein, H. M. Taylor, Smith and Bryant). Smith and Bryant are not
correct in saying (p. 142) that they cannot be traced further back than
lardnei^s Euclid (1828); they are to be found in Gregory of St Vincent's
work (published in 1647) Opus geometricum quadraturae drcuU ei secHanum
com. Book 1. Pt a, Props. 44, 45 (pp. 31, 32).

To prove 11. 12, take an obtuse-angled triangle ABC in which the angle at
^ \8 the obtuse angle.

1

J

II. 12]

PROPOSITION 12

40s

Describe squares on BCy CA, AB, as BCED, CAGF, ABKH.

Draw ALy BMy CN^ perpendicular to BC^ CA^ AB (produced if neces-
sary), and produce them to meet the further
sides of the squares on them va P^ Q^ R re-
spectively.

Join ADy CK.

Then, as in i. 47, the triangles KBCy ABD
are equal in all respects ;
therefore their doubles, the parallelograms in
the same parallels respectively, are equal ;

that is, the rectangle BP is equal to the
rectangle BR.

Similarly the rectangle CP is equal to the
rectangle CQ.

Also, if BGy CH be joined, we see that
the triangles BAG^ HAC are equal in
all respects ;
therefore their doubles, the rectangles AQ^ AR^ are equal.

Now the square on BC is equal to the sum of the rectangles BP^ CP^

i.e. to the sum of the rectangles BR^ CQ^

i.e. to the sum of the squares Blf, CG and

the rectangles AR^ AQ.

But the rectangles AR^ AQ axe equal, and they are respectively the
rectangle contained by BA^ AJVand the rectangle contained by CA^ AM.

Therefore the square on BC is equal to the squares on BA^ AC together
with twice the rectangle BA^ A Nor CA^ AM,

Incidentally this proof shows that the rectangle BA^ AN is equal to the
rectangle CA^ AM: a result which will be seen later on to be a particular
case of the theorem in in. 35.

Heron (in an-NairlzI, ed. Curtze, p. 109) gives a "converse" of 11. 12
related to it as i. 48 is related to i. 47.

In any triangle^ if the square on one of the sides is greater than the squares
on the other two sideSy the angle caniaitud by the latter is obtuse.

Let ABC be a triangle such that the square on BC is greater than the
squares on BA^ AC.

Draw AD at right angles to AC and
of length equal to AB.

Join DC.

Then, since DA C is a right angle,
the square on DC is equal to the squares
on DAy ACy [1. 47]

i.e. to the squares on BA^ AC.

But the square on BC is greater than
the squares on BAy AC; therefore the square on BC is greater than the
square on DC.

Therefore BC is greater than DC.

Thus, in the triangles BAC, DAQ
the two sides BA^ ACue equal to the two sides DA^ ^C respectively,
but the base BC is greater than the base DC.

4o6 BOOK II [n. i2» 13

Therefore the angle BAC is greater than the angle DAC\ [i. 25]

that is, the angle BA C is obtuse.

Proposition 13.

In acute-angled triangles the square on the side subtending
the acute angle is less than the squares on the sides containing
the acute angle by twice the rectangle contained by one of the
sides about the acute angle, namely thcU on which the per-
pendicular fcUls, and the straight line cut off within by the
perpendicular towards the acute angle.

Let ABC be an acute-angled triangle having the angle
at B acute, and let AD be drawn from the point A perpen-
dicular to BC ;

I say that the square on y^C is less than the squares on
CB, BA by twice the rectangle contained
by CB, BD.

For, since the straight line CB has
been cut at random at D,

the squares on CB, BD are equal to
twice the rectangle contained by CB, BD
and the square on DC. [11. 7]

Let the square on DA be added to
each;

therefore the squares on CB, BD, DA are equal to twice
the rectangle contained by CB, BD and the squares on AD,
DC.

But the square on AB is equal to the squares on BD,
DA, for the angle at D is right ; [i. 47]

and the square on -^C is equal to the squares on AD, DC\
therefore the squares on CB, BA are equal to the square on
AC and twice the rectangle CB, BD,

so that the square on ^C alone is less than the squares
on CB, BA by twice the rectangle contained by CB, BD.

Therefore etc.

Q. E. D.

As the text stands, this proposition is unequivocally enunciated of actae-
an^ed triangles ; and, as if to obviate any doubt as to whether the restriction
was fiilly intended, the enunciation speaks of the rectanp^le contained by one
of the sides containing the acute amgle and the straidit line interoepted
within by the perpendicular towards die acute angle. On the other hand, it

;

J

r

"• 13]

PROPOSITIONS 12, 13

407

is curious that it speaks of the square on the side subtending the acute angle;
and again the setting-out begins " let ABC be an acute-angl^ triangle haHng
the angle at B acute^^ though the last words have no point if all the angles of
the triangle are necessarily acute.

It was however very early noticed, not only by Isaacus Monachus,
Campanus, Peletarius, Clavius, Commandinus and the rest, but by the Greek
scholiast (Heiberg, Vol v. p. 253), that the relation between the sides of a
triangle established by this theorem is true of the side opposite to, and the
sides about, an acute angle respectively in any sort of triangle whether acute-
angled, right-angled or obtuse-angled. The scholiast tries to explain away the
word "acute-angled" in the enunciation: "Since in the definitions he calls
acute-angled the triangle which has three acute angles, you must know that he
does not mean that here, but calls all triangles acute-angled because all have
an acute angle, one at least, if not all. The enunciation therefore is : 'In any
triangle the square on the side subtending the acute angle is less than the
squares on the sides containing the acute angle by twice the rectangle, etc' "

We may judge too by Heron's enunciation of his "converse" of the
proposition that he would have left the word "acute-angled" out of the
enunciation. His converse is : In any triangle in which the square on one of
the sides is less than the squares on the other two sides^ the angle contained by the
latter sides is act^e.

If the triangle that we take is a right-angled triangle, and the perpendicular
is drawn, not from the right angle, but from the acute angle
not referred to in the enunciation, the proposition reduces
to I. 47, and this case need not detain us.

The other cases can be proved, like 11. 12, after the
manner of i. 47.

Let us take first the case inhere all the angles of the
triangle are acute.

H

i

D P E

As before, if we draw ALP, BMQ, CNR perpendicular to BCy CA, AB
and meeting the further sides of the squares on BC, CA, AB in P, Q, P, and
if we join ^C, AD, we have

the triangles I^BC, ABD equal in all respects,
and consequently the rectangles BP, BR equal to one another.
Similarly the rectangles CP, CQ are equal to one another.

4o8

BOOK II

[n.13

Next, by joining BG^ CH^ we prove in like manner that the rectangles AR^
i^Q are equal

Now the square on BC is equal to the sum of the rectangles BP^ CP^

Le. to the sum of the rectangles BR^ CQ^

Le. to the sum of the squares Blf^ CG diminished by the rectangles
AR.AQ.

But the rectangles AR^ AQ sue equal, and they are respectively the
rectangles contained by BA^ AN and by CA^ AM.

Therefore the square on BC is less than the squares on BA^ AC by
twice the rectangle BA^ AN or C4, AM.

Next suppose that we have to prove the theorem in the case where the
triangle has an obtuse angle at A.

Take B as the acute angle under considera-
tion, so that ^C is the side opposite to it

Now the square on CA is equal to the
difference of the rectangles CQ, AQ^

i.e. to the difference between CP and

AQ,
Le. to the difference between the square
BE and the sum of the rectangles
BP, AQ,
Le. to the difference between the square
BE and the sum of the rectangles
BP, AR,
Le. to the difference between the sum of
the squares BE^ BH and the sum
of the rectangles BP, BR

(since AR is the difference between BR and BET^

But BP^ BR are equal, and they are respectively the rectangles CB^ BL
and AB, BN.

Therefore the square on CA is less than the squares on AB^ BC by twice
the rectangle CB, BL or AB, BN.

Heron's proof of his converse proposition (an-Nairizi, ed. Curtze, p. no),
which is also piven by the Greek scholiast above quoted,
is of course simple. For let ABC be a triangle in which
the square on AC is less than the squares on AB^ BC.

Draw BD at right angles to BC and of length equal
XoBA.

Join DC.

Then, since the angle CBD is right,
the square on DC is equal to the squares on DB^ BC^
Le. to the squares on AB^ BC. [i. 47]

But the square on ^C is less than the squares on
AB, BC.

Therefore the square on ^C is less than the square on DC.

Therefore -^C is less than DC.
Hence in the two triangles DBC^ ABC the sides about the angles DBC^
ABCut respectively eqwO, but the base DC is greater than the base AC.

?

II. 13, 14] PROPOSITIONS 13, 14 409

Therefore the angle DBC (a right angle) is greater than the angle ABC
[i- 25]) which latter is therefore atute.

It may be noted, lastly, that 11. 12, 13 are supplementary to i. 47 and
complete the theory of the relations between the squares on the sides of any
triangle, whether right-angled or not.

Proposition 14.

To construct a square equal to a given rectilineal figure.

Let A be the given rectilineal figure ;
thus it is required to construct a square equal to the rectilineal

^ figure A.

5 For let there be constructed the rectangular parallelogram
BD equal to the rectilineal figure A. [i. 45]

Then, if BE is equal to ED, that which was enjoined
will have been done ; for a square BD has been constructed
equal to the rectilineal figure A.
10 But, if not, one of the straight lines BE, ED is greater.
Let BE be greater, and let it be produced to F\
let EF be made equal to ED, and let BF be bisected at G.

With centre G and distance one of the straight lines GB,
GF let the semicircle BHF be described ; let DE be produced
15 to H, and let GH be joined.

Then, since the straight line BF has been cut into equal
segments at G, and into unequal segments at E,

the rectangle contained by BE, EF together with the
square on EG is equal to the square on GF. [11. s]

so But GFv& equal to GH\
therefore the rectangle BE, EF together with the square on
GE is equal to the square on GH.

But the squares on HE, EG are equal to the square on
GH, [1.47]

25 therefore the rectangle BE, EF together with the square on
GE is equal to the squares on HE, EG.

Let the square on GE be subtracted from each ;

4IO BOOK II [iL 14

therefore the rectangle contained by BE, EF which
remains is equal to the square on EH.
p But the rectangle BE, EF is BD, for EF is equal to ED ;

therefore the parallelogram BD is equal to the square on
HE.

And BD is equal to the rectilineal figure A.

Therefore the rectilineal figure A is also equal to the square
[5 which can be described on EH.

Therefore a square, namely that which can be described
on EH^ has been constructed equal to the given rectilineal
figure A. Q. E. F.

7. that which was enjoined will have been done, literally **woald have been
done," TffTorif Am ^ r^ HiraxOiw*

35> 36. which can be described, expressed by the future passive participle, i^vypa^

Heiberg (Mathemaiisches zu Arisioteies, p. 20) quotes as bearing on this
proposition Aristotle's remark {De anima 11. 2, 413 a 19: of. Metaph. 996 b 21)
that "squaring^ (rcrpaymvurfuk) is better defined as the ''finding of the mean
(proportional) than as ''the making of an equilateral rectangle equal to a
given oblong," because the former definition states the cause^ the latter the
condumn only. This, Heiberg thinks, implies that in the text-books whidi were
in Aristotle's hands the problem of n. 14 was solved by means of proportions.
As a matter of &ct, the actual construction is the same in 11. 14 as in vi. 13 ;
and the change made by Euclid must have been confined to substituting in
the proof of the correctness of the construction an argument based on the
principles of Books i. and 11. instead of Book vi.

As n. 12, 13 are supplementary to i. 47, so 11. 14 completes the theory of
transformation of areas so far as it can be carried without the use of proportions.
As we have seen, the propositions i. 42, 44, 45 enable us to construct a
paraUelogram having a given side and angle, and equal to any given rectilineal
figure. The parallelogram can also be transformed into an equal triangle with
the same given side and angle by making the other side about the ang^ twice
the length. Thus we can, as a pardcidar case, construct a rectangle on a
given iMLse (or a right-angled triangle with one of the sides about me right
angle of given length) equal to a given square. Further, L 47 enables us
to make a square equal to the sum of any number of squares or to the
difference between any two squares. The problem still remaining unsolved is
to transform any rectangle (as representing an area ec^ual to that of any
rectilineal figure) into a square of equal area. The solution of this problem,
given in II. 14, is of course the equivalent of the extraction of the square root,
or of the solution of the pure quadratic equation

o^^ab.

Simson pointed out that, in the construction given by Euclid in this case,
it was not necessary to put in the words **i>/ BE be greater,** since the
construction is not affected by the question whether BE or ED is the greater.
This is true, but after all the words do little harm, and perhaps Eu^d may
have regarded it as conducive to clearness to have the points B, G, £, JFin
the same relative positions as the corresponding points A, Q D, B in the
figure of II. 5 which he quotes in the proof.

INDEX OF GREEK WORDS AND FORMS.

dyc^tor, angU'Uss (figure) 187

Mwarw : ^ e/f rh d^. iLwaytaytit ^ dcA roG d5.

deZftr, ^ e^f rh dd. ^Toutf-a dv6dci(tf 136
iKibo€iiiit, hoHhlike 188
d/i/9Xeia (7wr(a), obtuse (angle) 181
d/ijSXv7i^i0f, obtuse-angled 187
d/np^, indivisible 41, 168
d/A^KoiXos (of curvilineal angles) 178
dfiipUvpTot 178
d^aypd^uf dv6 to describe on contrasted with

to construct (owrHi99cdfu) 548
dFf£Kv6iax9os (r^vof), Treasury cf Analysis 8,

10, II, 138
iporrpo^KAt (species of locus) 3^0
dwofiMOfup^, nan-uniform 40^ lOi-s
drriffrpo^t conversion 356-7 : i!fA/i^ variety,

^ Tpottyov/idrn or ^ KVpUtt, ibid,
dw&wapicT09f non-existent 119
ib/MToSt indeterminate : (of lines or curves)

160: (of problems) 199
dvaTwyi^, reduction 135 : c^r rh dddraror 136
dvecpof, infinite: ^ ^ dv. ^ff/9dX\ofi^ny of

line or curve extending without limit and

not ** forming a figure 160-1 : hr* dv. or

c/r dv. adverbial 190: ht* dv. 5iaf/>c£^at

968: Aristotle on rb Aweipw 133-4
dvXoCt, simple : (of lines or curves) 161-1 :

(of surfiices) 170
d«'6<ei(cr, proof (one of necessary divisions of

a proposition) 139, 130
UrrwBai^ to meet (occarionally touch) 57
Apfnrroti irrational : of Xbyw 137 : of diameter

(diagonal) 399
iiov/iparog, mcompatible 139
d^Aurr«»rot, not-meeting, non-secant, asymp-
totic 40, 161, 303
iabi^Btrotf incompodte: (of lines) 160, 161 :

(of surfisces) 170
drcurrof, unordered: (of problems) 138: (of

irrationals) ri5
droftoi ypa/ifud, "indivisible lines'* 368
d^= segment of circle less than semicircle

187
fidBos, depth 158-9
^Id^tr, base 348-9
yrfpi^u 343
yvibfuuf, tu gnomon
ypofi/iij, line (or curve) ^.v.
ypofifwcQtt graphically 400
itbofUwot given^ different senses 133-3:

Euclid's de8o/A(fra or Daia^ q,v.
My/tara^ iltustrationst of Stoics 339
dci ^, "thus it is lequired," introducing

hopia/ibt 393
dcdypofifias proposition (Aristotle) 353
lia^N^iff: point ^division (Aristotle) 165,

170, 171: method of division (exhaustion)

385 : Endid's wtpl buup^euo 8, 9, 18, 87, 1 10
Bmurrdatu, almoatssdimensions 157, 158
liatfTar^r extended, 4^* H one way, iwl dbo

two ways, ivi roUk tkrm ways (of lines,

foi&ces and aolkss reflectively) 158, 170

<id(rn;fia, distance 166, 167, 307 : (of radius of
circle) 1 99 : (ofan angle) = divergence 1 76-7

bie^obucbt (of a class of loci) 330

bvtfxOto, "let it be drawn through" (= pro-
duced) 380

biopiffti.bt—{i) particular statement or defini-
tion, one of the formal divisions of a pro-
position 139: (3) statement of condition of
possibility 138, 139, 130, 131, 334, 343, 393

doayioy^ bpftMrnHj, IntroductMn to Harmony,
by Cleonides 17

Uarip^ ixaHp^t meaning respectivdy 348, 350

igfitfiMjadioatiM, use of, 344

^irc&of = Euclid 400

0K$€ois, setting-out, one of formal divisions of
proposition 139: may be omitted some-
times 130

iirrbs, Karb rb (of an exterior angle in sense
of re-entrant) 363 : ^ imbt ywpla, the
exterior angle 380

i\ucou8ijst spiral-shaped 159

AXet^if, fatling'Short (wiu reference to
application of areas) 36, 343-5* 3^3-4

cXXiv^r vpbfiKfifUL, a deficient (= indeter-
minate) problem 139

iwaXKd^, tdtematdy or (adjectivally) alternate
308

Irroca, notion, use of, 331

IrffToo'tr, objection 135

^rr6r, xarb rb or ^ hrbs {v^^) of an interior
angle 363, 380: ^ ii^bt koI dvcyorrW
Twria, the interior and opposite angle 380

irtiubxiw^oM {hri^Hbywufu, join) 343

iwlvtbop, plane in Euclid, used for smfaee
also in Plato and Aristotle 169

iwtwpoo0€tp, MTp99$ep eZrot, to stand in
front of (hiding from view) 165, 166

hri^djftia, surface (Euclid) 169

Hepb/iffKes, oblong 151, 188

€b$i6, rb, the straight 159: tbd^ (Ypo/ifii}),
straight line 165-9

tbObypatifiot, rectilineal 187: neuter as sub-
stantive 346

4^wTta$ai, to to$teA 57

l^op^eir, to coincide, 4^ap/ib^a$tu, to be
applied to 168, 334-5, ^^

i^KTucbs (of a class of loci) 330

i^^ift, "in order" 181: of adjacent angles
181, 378

Btib/nifjM, theorem f.v»

0vp€bt (shield) sdhpse 165

twwQv WSiy (horse-fetter), name for a certain
curve 163-3, 176

iaofiirpioif 9x^i»Aruio, repL, on isometric figures
(Zenodoms) 36, 37, 333

KdBrros €b$€ia ypatiti-fj, perpendicular 181-3,
371: "plane" and '^sofid" 373

Ko/iwbXot, curved (of lines) 150

KamoKwii, construction, or macainery, one of
divisions of a proposition 139: sometimes
unnecetiary 130

Ktemrikii mvireti SecOo cauamis, of Endid 17

412

INDEX OF GREEK WORDS AND FORMS

KfMv, **let it be made'* t6g
K€Ka/M4iihil, bent (of lines) 159, 176
Kirrpoff centre i83» 184, 199 : iiix rod Kirrpov

s radios 109
ffcparoct^ (fitwUk)^ kom-Hki (angle) 177,

178, 189
irXfir, to inJUct or dtfiecty rtcXda^cu, n^haa-

fUinit Kkins 118, 150, 159, 176, 178
KKtff It t inclutoHan^ 176
cocXoTi^tor, koiiauhangUd figure (Zenodorus)

«7, 188
cmmU tnfouut Common Notions (s axioms)

111-9: called also rd mpd, cocmU 96^ai

(Aristotle) 110, 111
Koutif wpoaK€lff$t, i^jfprfja0» 176
KOffv^t vertex : *ard copu^r, tftrticai (angles)

178
»plu9tt ring (Heron) 16^
X%&/ia, lemma (s something assumed, Xa/bi-

(ivhiuifm) 133-4
X0iv6ff : Xmv^ ^ AA Xmv j rf BH Cinf ^«rr(r 145
liifni^ parts (s direction) 190, 308, 313 :

(=side) 171
M^Kot, length, 158-9
fMfrocc^, luHt'likg (of angle) 16, 101 : ih

/iipoei^ff (^xfMA), lune 187
/UKT^, '* mixed' (of lines or curves) 161, 161 :

(of surfiices) 170
/uordff wpovXafiofha tf/tf'&v, definition of a pcmi

155
fi«^6rrp9^ IXi( '* single-turn spiral" 111-
311., 164-5: in Pappus s cylindrical helix

9t6ff€Uf ifuiittaiwm, a class of problems
150-1: rctffty, to verge 118, 150

^vcTpaeM^t scraper'Uke (of angle) 178

d^oec^, '*of the same form " 150

0A&ocot, "similar" (of numbers) 357 : (of angles)
s equal (Thales, Aristotle) 151

6iMioiuf/fi% uniform (of lines or curves) 40,
161-1

4(eca (7wr(a), acute (angle) 181

d^vyiirtot, acute-angled 187

tw€p Hu det^ (or voc^ot) Q.B.D. (or P.) 5j^

6p$vpiiinoSt right-angled : as used of quadn-
laterals =f)f^aM^ar 188-^

dpor, 6piaMot^ definition 143 : original mean-
ing of Spot 143: = boundary, limit 181

S^tf visual ray 166

wd^Tjf |irraXa/i5ar6/u€rat, " taken together in
any manner" 181

wapafioMi two x**P^9 applUatum of areas 36,
343-5: contrasted with Wtp^ii (exceed-
inj^ and AXci^ct (falUng^skort) 343 : rapa-
fioKii contrasted with o^araait (construetion)
343: application of terms to conies by
ApoUonius 344-5

ropddofot rdroi, A, **the Treasury of Para-
doxes" 319

weLpoKKdrrw, •*fall beside" or "awry" 161

vopavXi^/w/ia, complenunt^ q*v.

w4p(u, extremity 165, 181: Hpas ovyKktUo
(Potidonius' definition oifigur^ 183

wtpuxo/juhri (of angle), Ttpux^f^^^o (of rect-
angle), coHtamea 370: rh dlt rcptcx^^ror,

twice Hit rectangle contained 380: (of figure)

contained or bounded 181, 183, 184, 180, 187
w€pt^peuL, circumference 184
T€pu^p^ftt circular 15^
vcpc^cp^pofifiof, contained by adrcumference

of a circle or by arcs of circles 181, 184
vXirof, breadth 158-9
wXeood^oo (vp6/9Xva), "(problem) in excess**

119
vAXor, a mathematical instrument 370
wciKAwXevpo^t manv-sided figure 187
woplffooBai, to **find" or "fitrnish" 115
wiptciuL^ porism q.v,
wpdpXfipa, problem g.v,
wporrtod/uoos, ieading". (of conversion) scorn-

Sete 356-7 : wptnrYod/upoo (Bmipti/io) kaelit^g
leorem) contrasted with converse 157
vp^t, in geometry, various meanings 177
wpiroiott, emmaaHom 119-30
wportlov^ ** propound" 118
rpt^off, prime, two senses of, 146
«TA0>if, eau 134
^6r, rational 137: fifHi hAt^erpat r%i rcfi-

rd^ (** rational diaineter of 5") 399
nffMcbr, point 155-6
ord9paif a mathematical instrument 371
OTtypkii, point 156
rmxttoo^ element 114-6
OTpoyyiiKoOf rd, round (circular), in Plato

159, 184: orpoyyvKAnif, roundness 181
^v/iwipaofui, conclusion (of a proposition)

119* 130
o^Brrot, composite : (of lines or curvet) 160 :

(of surfaces) 170
^dreitftf, convergence 181
owloTotoBait construct', special connotation

159, 180 : with hnh^ 189: contrasted with

wapafid>Ji€tP (apply) 343
axi7MaT07pa^cy, ox^itaroypoi^U^ represent-
ing (numbers) by figures of like shape 359
^XVMAToroioOtf'a or oj^pm vocoGff'a, forming a

figure (of a line or curve) 160-1
rmypipoo (of a problem), "ordered" 118
rerpaytootffpift squeui^g^ definitions of 149-

50, 410
rmdywror, square: sometimes (but not in

Euclid) any four-an^ed figure 188
rtrpiar\€vpo9^ quadrilateral 187
ropiii^ section, s/mw/ ^section 170, 171, 178
rowucho Beibpnpot hcus^tkeorem 319
rhfwoti locus 319-31: room or naoe i3it.:

place (where things may be round), thus

T^ot6ooXv6pi£ootS, 10 : ropddo^t rdrot 319
rSpoot, instrument for drawing a circle 371
rplw\9vpoo, three-sided figure 187
rvx^ nffMcbr, a point at rtmdom 151
inrtpfioMi, exceedinf^ with reference to method

of application of areas 36, 343-5, 386-7
6r6, in expressions for an aiwle (ii^ Sw6 BAF

yuolo) 149, and a rectangfe 370
^teetroft, "is by hypothesis" 303, 311
inror^oof^ suHend, with ace. or m and ace

«49' «83, 350
^ptepuhni ypapLf^ ditenmsuOe line (curve),

**forming a figure" 160

J

ENGLISH INDEX.

:

I

al.'Abb&s b. Said al-Jauharl 85

^^Abthiniathns" (or **Anthisathus'*) 103

Abu '1 *Abb&s al-Fadl b. Hfttim, see an-
Nairid

Abu * Abdallih Muh. b. Mu*ftdh al-Jayyftnl 90

Abu 'All al-BasrI 88

Aba 'All al-Hasan b. al-Hasan b. al-Haitham
88,89

Abu D&'ud Sulaimftn b. *Uqba 85, 90

Abu Jafar al-Khftzin 77, 85

Abu Ja*far Muh. b. Muh. b. al-l^asan
Na^ddln at-XOsI, sa Naflraddin

Abu Muh. b. Abdalbftql al-Ba^dAdl al-Fara^i
8it., 90

Abu Muh. al-Hasan b. *UbaidalUh b. Sulai-
man b. Wahb 87

Abu Nafr 6an al-Na*ma 90

Abu Nafr Mansur b. *A1I b. *Ir&q 90

Aba Na$r Muh. b. Mul^. b. Tarkhin b.
Uzlag al-F&rib! 88

Abu Sahl Wljan b. Rustam al-Kuhl 88

Abu Said Sinin b. Thftbit b. Qurra 88

Abu 'UthmiD ad-Dimashql 95, 77

Abu '1 Wafa al-BQzjftnl 77, 85, 86

Abu Yusttf Ya qub b. Ithiq b. af -Sabbih al-
KiDd! 86

Abu Yusuf Ya qub b. Muh. ar-RAd 86

Adjacent (^^(^), meaning 181

Aenaeas (or Aigeias) of Hierapolis 18, 311

Aganis 17-8, 191

Ahmad b. al-Husain al-AhwftzI al-KAtib 89

Ahmad b. *Umar al-KariblsI 85

al-Ahwftzi 89

Aigeias (? Aenaeas) of Hierapolis 38, 311

Alexander Aphrodisiensis 711., 39

Algebra, geometrical, 371-4 : classical method
was that of £ucl. II. (cf. ApoUonius) 373 :
preferable to semi-algebraidd method 377-
8: semi-algebraical method due to Heron
373, and favoured by Pappus 373 : geome-
trical equivalents of algebraical operations
374 : algebraical equiviuents of propositions
in Book 11. 373-3

'All b. Ahmad Abu 1 Qftsim al-AntikI 86

Allman, G. J. 13511., V^% 35*

Alternate (angles) 308

Alternative proofs, interpolated, 58, 59

Amaldi 175, i79-*>. «93. «oi, 313, 338

Ambiguous case 306-7

Amplunomus 135, 138, 15011.

Amjdas of Heradea 117

Analysis (and synthesis) 18: alternative
proofs of XIII. 1-5 by, 137: definitions o(

interpolated, 138: described by Pappus
138-9: modem studies of Greek analysis
1 30 : theoretical and problematical analysis
138: TVeasury 0/ analysis (r^irot hf^Lhth-
/lOfot) 8, 10, If, 138: method of analysis
and precautions necessary to 139-40:
analysis and synthesis of problems 140-3 :
two parts of analysis (a) transformation^
CctS resolution^ and two parts of ^thesis,
(a) construction f (b) demonstration 141 :
example from Pappus 141-3: analysis
should also reveal oiopiffiiMt (conditions of
possibility) 143

Anal3rtical method 36: supposed discovery
of, by Plato^i34, 137

Anaximander 370

Anchor-ring 163

Andron 136

Angle. Cnrvilineal and rectilineal, Euclid's
oeiinition of, I76sq.: definition criticised
by Syrianus 176: Aristotle's notion of
emgie as nXioit 176: ApoUonius' view of,
as contraction 176, 177 : Plutarch and
Carpus on, 177 : to which category does it
belong? qucmtum^ Plutarch, Carpus, "A-
ganis" 177, Euclid 178; fuale^ Aristotle
and Eudemus 177-8: reUUton^ Euclid 178 :
Syrianus* compromise 178 : treatise on the
Anfie by Eudemus 34, 38, 177-8: classifi-
cation of angles (Greminus) 178-9: cnrvi-
lineal and "mixed" angles 36, 178-9,
hom-Uki (ircparoeidth) I77t 178, 183, 365,
tune-tike (/iiyi^oei^f) 36, 178-9, scraper'tike
{ivorponi-fp) 178 : angle </a segment 353 :
ang^e^a semicircle 183, 353: definitions
of ai^le classified 170 : recent Italian views
179-81: angle as cluster of straight lines
or rays 180-1, defined by Veronese 180:
as part of a plane (''angular sector") 179-
ioijlat angle (Veronese etc.) 180-1, 369:
three kinds of angles, which is prior
(Aristotle)? 181-3: adjacent angles 181:
alternate 308 : similar (= equal) 178, 183,
353: vortical 378: exterior and interior
(to a figure) 363, 380: exterior when re-
entrant 363: interior and opposite 380:
constraction by ApoUonius of angle equal
to angle 396 : angle in a semicircle, theorem
of, 31 7-19 : trisection of angle, by conchoid
of Nicomedes 365-6, by ouadratrix of
Hippias 366, by spiral of Archimedes 367

al-AntikI 86

Antiplion 7*., 35

414

ENGLISH INDEX

•^Anthisathus" (or "Abthiniathus") «03

Apastamba-^ulba-Sutra 351 : evidence in, as
to early discovery of Eucl. i. 47 and use
of pomon 360-4: BUrk*s claim that
Indians had discovered the irrational 363-
4: approximation to ^2 and Thibaut's
explanation 361, 363-4: inaccurate values
of r in, 364

Apollodonu "Logisticus" 37, 319, 351

Apollonius: disparaged by Pappus in com-
parison with EucGd 3: supposed by some
Arabians to be author of the EUnunis 5 :
a ''carpenter'* 5 : on elementary geometry
43: on the lin€ 150: on the tmgU 170:
general definition oxdiamtUr 315 : tried to
prove axioms 41, 69, 391-3 : his "general
treatise" 43: constructions by, for bisec-
tion of straight line 368, for a perpendicular
470, for an angle equal to an angle 3^:
on parallel-axiom (?) 43-3: adaptation to
comes of theory of application of areas
344-5 : geometrical algebra in, 373 : Plane'
^^* t4« *59f 330: PlamvtOffta 151 : com-
parison of dodecahedron and icosahedron
6 : on the cochUas 34, 43, 163 : on unordered
irrationals 43, 115: 138, 188, 331, 333, 346,

^ «59» 370, 373

Application of areas 36, 341-5 : contrasted
with txceeding and faUit^short 343 :
complete method equivalent to geometric
solution of mixed quadratic equation 344-51
383-5, ^86-8 : adaptation to conies (Apol-
lonius) 344-5 : apkicatim contrasted with
€onstru€tion (Produs) 343

"Aaaton" 88

Arabian editors and commentators 75-90

Arabic numerals in scholia to Book x.,
I3thc., 71

Archimedes 116, 143: "postulates" in, i30,
133: fSunous ** lemma ' ' (assumption) known
as Postulate of Archimedes 334: '* Porisms"
in, 1 1 »., 13 : spiral of, 36, 367 : on straight
lifui66: orkptane iTi-ii 335,370

Archytas 30

Areskong, M. E. 113

Arethas, Bishop of Caesarea 48: owned
Bodleian MS. (B) 47-8: had famous Plato
MS. of Patmos (Cod. Clarkianus) written 48

Argyrus, Isaak 74

Anstaeus 138: on conies 3: Solid Loci 16,
439: comparison of five (regular solid)
figures 6

Aristotelian Problems 166, 183, 187

Aristotle: on nature of elements 116: on
first principles 1 1 7 sqq. : on definitions 1 1 7,
119-30, 143-4, 146-50: on distinction be-
tween hjrpotheses and definitions 1 19, 130,
between hypotheses and postulates ii8»
119, between hypotheses and axioms i3o:
on axioms 11^31: axioms indemon-
strable 131 : on definition by negation
156-y: on poisUs 155-6, 165: on lina^
defimtions of 158-9, classification of 155^

• 60: quotes Plato's definition of ttrasght
line 166: on definitions oi surfaa 170:

on the angle 176-^ : on priority as between
right and acute angles 181-3 : on Jigure
and definition of 183-3: ddKnitions ik
"squaring" 149-50, 410: on parallels 190-
3, 508-9: aa gnomon 351. 355, 359: on
attnlmtes card wtutrbt and wpChvo KuBiikov
3>9> 3^^ 3^5: on the objection 135: on
reduction 135: on reduetio ad absurdum
136: on the infinite 333-4: supposed pos-
tulate or axiom about divergent fines taken
by Produs firom, 45, 307 : gives pre- Eucli-
dean proof of I. 5 353-3: on tneorem of
angle m a semicircle 149 : on sum of an^es
of triangle 319-31 : on stun of exterior
angles orpolvgonj33: 38, 45, 117, 15011.,

l8f, 184, 185, 187, 188, 105, t03, 303f
331, 333, 333, 330, 359, 963-3, 983

al-Aijftnl, Ibn RAhawaihi 86

Ashkil at-U'sIs5is.

Ashraf Shamsaddin as-Samarqandl, Mu^. b.
5«., 89

Astaroff, Ivan 113

Asvmptotic (non-secant) lines 40, 161, 303

Athelhard of Bath 78, 93-6

Atfaenaens of Cyzicns 117

August, E. F. 103

Austin, W. 103, III

Autc^ycus, On the moving sphere 17

Avicenna 77, 89

Axioms, distinguished from postulates by
Aristotle 1 18-9, by Ptodus (Geminus and
"others"} 40, 131-3: Produs on diflS-
cnlties in distinctions 133-4: distinguished
from hypotheses, by Aristotle 130-1, by
Produs 13 1-3: indemonstnible I3i: at-
tempt by Apollonius to prove 993-3:
=s" common (thii^)" or **oommon
opinions" in Aristotk I30, 33 f: common
to all sdences 119, 130: odled "common
notions*' in Euclid i3i, 331: which are
genuine? 331 sqq. : Produs recognises five

333, Heron three 333 : interpolated axioms

334, 333: Pappus' additions to axioms
35, 333, 334, 333 : axioms of congraenoe,
(i) Eudid's Common Notion 4, 394-71
(3) modem systems (Pasch, Veronese and
Hilbert) 338-31: **axiom" with Stoicss
every simple declaratory statement 41, 991

Babylonians, knowledge of triangle 3, 4, 5,

Bacon, Roger 94

Balbns, de mensnris 91

Barbann 319

BarUam, arithmetical commentary on End. I|.

„74

Barrow 103, 105, no, in

BasCf meaning 348-9

Basd, editio prineeps of EucL loo-i

Basilides of Tyre 5, 6

Bftudhiyana Sulba-S&tra 360

Bayfius (Balf, Laaure) 100

Becker, J, K. 174

Bees 176

Bdtrainiy £. 319

'

.A

J

.^t

ENGLISH INDEX

41S

Benjamin of Lesbos 113

Bergfa, P. 400-1

Bernard, Edward 101

Besthorn and Heiberg, edition of al-Hajjaj*s

translation and an-NairIzI*s commentary

««, 91 n., 7911.
Bhftskara 355

BiUingsley, Sir Henry 109-10
al-BlrOnl 90

Bjombo, Axel Antfaon 1711., 93
Boccaccio 96
Bodleian MS. (B) 47, 48
Bocckh 351, 371
Boethius 99, 05, 184
■ Bologna MS. (b) 49

k Bolyai, T. 119

Bolyai, W. 174-5, «I9» 3«8

Bolzano 167

Boncompagni 931V., 104*.

Bonola, K. 401, 319, 337

Borelli, Giacomo Alfonso 106, 194

Boundary (Upot) 182, 183

Br&kenhjelm, P. R. 113

Breitkopf, Job. Gottlieb Immanael 97

Bretschneider 13611., I37*S95> 304. 344. 354»

358
Briconnet, Fran9ois 100
Briggs, Henry 101

BriL Mus. palimpsest, 7th— 8th c, 50
Bryson Sm.
Biirk, A. 35*, 360-4
Btirkien 179
Buteo (Borrel), Johannes 104

Cabasilas, Nicolans and Theodonis 73

Caiani, Angdo 101

Camerarius, Joachim loi

Camerer, J. G. 103, 493

Camorano, Rodrigo lit

Campanus, Johannes 3, 78, 94-96, 104, 106,

no, 407
Candalla, Franciscns Flussates (Fran9ois de

Foix, Comte de Candale) 3, 104, no
Cantor, Moritx 711., to, «7«, 304, 318, 310,

333. 35»» 355» 357-8. 360, 401
Carduchi, L. iis
Carpus, on Astronomy 34, 43: 45, 197, 118,

177
Case, technical term 134: cases interpolated

Casin 411., 919.

Cassiodorius, Magnus Aurelius 99

Cataldi, Pietro Antonio 106

CatoUrica, attributed to Euclid, probably
Theon's 17: Caiopiriea of Heron si, 953

"Cause": consideration of, omitted by com-
mentators 19, 45: definition should state
cause (Aristotle) 149: causes middle term

. (Aristotle) 1 49 : question whether geometry
should investigate cause (Geminus), 45,
15011.

Censorinus 91

Centre, u4mfw 184-5

Ceria ArisMetica 35

Chasles on Poritms of Euclid io» 11, 14, 15

Chinese, knowledge of triangle 3, 4, 5, 351 :
•*Tcheou pei" 355

Chrysippus 330

Cicero 91, 351

Circle: definition of, 183-^5: around, crpoy-
y(iKop (Plato) 184: ^ w€pi^p6ypatifMP
(Aristotle) 184: a plane figitrt 183-4:
centre of, 184-5: pole of, 185: bisected by
diameter (Thales) 185, (Saccheri) i^i-ix
intersections with straight line 137-8,
373-4, with another circle 338-40, i43-3f
«93-4

Circumference, W€pt^4p€ta 184

Cissoid 161, 164, 176, 330

Clairaut pS

Claymundus, Joan. loi

Clavius (Christoph SchlUssel) 103, 105, 1941

«3«. 38'. 39«. 407

Cleonides, Iniroductton to Harmony 17

Cochiias or cochlion (cylindrical heUx) 163

Codex Leidensis 399, i: 33, 3711., 7911.

Coets, Hendrik 109

Commandinus 4, io3, 103, 104-5, «>^* i<o»
III, 407: scholia included in translation
of Elements 73: edited (with Dee) De
difrisionibus 8, 9, 110

Commentators on Eucl. criticised by Proclus
«9. «6, 45

Common Notions : ~ axioms 63, 1 30- 1,331-3:
which are genuine ? 33 1 sq. : meaning and
appropriation of term 331 : called "axioms**
by Proclus 33 1

Complement, roparXi^pwAca: meaning of, 34 1 :
"about diameter" 341: not necessarily
parallelograms 341 : use for application of
areas 343-3

Composite, vin^Btrw, (of lines) 160, (of sur-
faces) 170

Conchoids 160-1, 365-6, 330

Conclusion, vviiHpa^iui : necessary part of a
proposition 139^-30: particular conclusion
immediatel]^ made general 131 : definition
merely stating conclusion 149

Congruence-Axioms or Postulates: Conmion
Notion 4 in Euclid 334-^ : modem systems
of (Pasch, Veronese, Hubert), 338-31

Congruence theorems for triangles, recapitula-
tion of, 305-6

Conies, of Euclid 3, 16: of Aristaeus 3, 16:
of Apollonius 3, r6: fundamental property
as proved by Apollonius equivalent to
Cartesian equation 344-5 : focus-directrix ^
property proved by Pappus 15

Constantinus Lascaris 3

Construct {owioTao$ai)i contrasted with
describe on 348, with i^y to 343: special
connotation 359, 389

Construction, arara#ffc^, one of formal di-
visions of a proposition 139: sometimes
nnnecessaiy 130: turns nominal into Ireal
• definition 146: mechanical, 151, 387

Continuity, Principle of, 33^ sc|., 3^3, 3731 394

C<9ii««rxfi0M, geometrical : distinct from /^^o/
956 : '* laiding" and partial varieties 356-7,
337

416

ENGLISH INDEX

Copernicus loi

Cordonis, Mattheus 97

Cratistus 133

Crelle, on ^t plane 173-4

Ctesibitts so, ai, 391*.

Omn, Samael 11 1

Cnitze, Maximilian, editor of an-NairizI 33,

78. 9«. 94t 96. 97 «•
Curvet, classification of: see line
Cylindrical helix 161, 163, 339, 330
Czecha, Jo. 113

Dasypodius (Ranchfuss), Conrad 73, io3
Data of Euclid 8, 133, 141, 385, 391
Deahna 174
Dechales, Claude Franfois MiUiet 106, 107,

108, no
Dedekind's Postulate, and applications 335-40
Dee, John 109, no: discovered De divisi-

9ntiut 8, 9
D^nUion^ in sense of *' closer statement"

(d(o/M^/i4t), one of formal divisions of a

proposition 139: may be unnecessary 130
Definitions : Aristotle on, 1 1 7, 1 19, i30, 1 43 :

a class of tluHs (Aristotle) i3o: distin-

Siished from hypotheses 1 19, but confused
erewith by Produs 131-3 : must be
assumed 11 7-9, but say nothing about
existmci (except in the case ot a few
primary things) 119, 143: terms for, l^t
and hpiffuM 143 : real imd naminai defi-
nitions (real = nominal plus (XMtulate or
proof). Mill anticipated by Aristotle, Sac-
cheri and Leibniz 143-5: Aristotle's re-

anirements in, 146-50, exceptions 148:
^ould state cause or middle term and be
genetic 149-50: Aristotle on unscientific
definitions (^ir/«^ wporipuv) 1^8-9: Euclid's
definitions agree generally with Aristotle's
doctrine 146: interpolated definitions 61,
63 : definiuons of technical terms in Aris-
totle and Heron, not in Euclid 150

De levi et ponckresOf tract 18

Demetrius Cydonius 73

Democrittts 38

De Morgan 346, 360, 360, 384, 391, 398, 300,

309. 3»3. 3«4. 3>5. 3^» 37^

Desargues 193

Describe om (4ra7pd0eiv dvo) contrasted with
construct 348

De Zolt 338

.DiageneU (9ui7cinof) 185

••Diagonal" numbers: su "Side-" and
''diagonal-" numbers

Z>MMi«f^(dMiM«Tpot), of cirdeor parallelogimm
185: as applietl to figures generally 33ft:
•• rational ^ and * • irrational ^ diameter of 5
(Plato) 399, taken from Pythagoreans 399-

4»

Dimensions (cf. SuM'Ytttf-ctr) 157, 158: Aris-
totle's view of, 158-9

Dinostratus 117, 366

Diodes 164

Diodonis 303

Diogenes Laertius 37, 305, 317, 351

Diophantus 86

Diarismus (dio^t^/i^t) cb (a) •'definitioo" or
"specification," a formal division of a
proposition 1 39 : (b) conditicm of possibility
138, determines how fiur soluticm possible
and in how many ways 130-1, 343: duh
rismi said to have been discovered by
Leon 116: revealed by aneUysis 143: in-
troduced by if I ^ 393 : first instances in
Elements 334, 393

Dippe 108

Direction, as primaiv notioii, ditcussfd 179:
direction-theory ot parallds 191-3

Distance^ StAmitta : » nuiius 199 : in Aristotle
has usual general sense and s dimension 199

Division (method of), Plato's 134

Divisions {offyures) by Eudid 8, 9: trans-
lated by Muhammad al-BagdidI 8 : found
(by Woepcke) in Arabic 9, and (by Dee)
in Latin translation 8, 9 : no

Dodgson, C. L. 194, 354, 361, 313

Dou, Jan Pieterszoon 108

Duhamel 139, 338

Egyptians, knowledgeof right-angled triangles

35«
Elements: pre-Eudidean Elements, by Hip-
pocrates of Chios, Leon 116, Theudius 117:
contributions to, by Eudoxus i, 37, Theae-
tetus I, ^7, Hermotimus of Colophon
117: Eudtd's^/zMMW/j, ultimate aims of 3,
1 1 5-6: commentators on i9-45» Produs
10, 39-45 and passim^ Heron 30-34, an-
Nairlzl 31-34, Porphyry 34, Pappus 34-
37, Simplidus 38, Aenaeas (Aigeias) 38:
Mss. of 46-51 : Theon's changes in text
54-58: means of comparing Theonine with
ante-Theonine text 51-53: imeipolations
before Theon*s time 58-03 : scholia 64-74 •
external sources throwing lipht on text.
Heron, Taurus, Sextus Empincus, Produs,

lamblichus 63-3: Arabic translations fi)
by al-Hajjftj 75, 76, 70, 80, 83-4, (3) by
Ifh&q and Tliibit b. Qurra 75-80, 83-4,

(3) Naftraddin at-TusI 77-80, 84: Hebrew
translation by M(»es b. Tibbon or Jakob
b. Machir 70 : Arabian versions compared
with Greek text 79-83, with one another
83, 84: translation by Boethius 93: old
translation of loth c. 93: translation by
Athdhard 93-6, Gherard of Cremona 93-4,
Campanus 94-6, 97-100 etc^ Zamberti
98-foo, Commancfinus 104-5: introduc-
tion into England, loth c, 95 : translation
by Billingsley 109-10: Greek texts, eititio
princeps 100- 1, Gr^ory's 103-3, Peyrard's
103, August's 10^, Heiberg's^iajitw: trans-
lations and editions generally 07-113: on
the nature of dements (Proclu^ 114-^1
(Menaechmus) 1 14, (Aristotle) 1 16: Produs
on advantages of Euclid's Elements 115:
immediate reco^ition of, 116: first prind-
ples of, definitions, postulates, common
notions (axioms) 117-34: technical terms
in connexion with, 135-43 : no definitions

^1

1
J

ENGLISH INDEX

417

>'

r

of such technical terms 150: sections of
Book I. 308

Elinuam 95

Eng^el and SUickel 319, 331

Enriques, F. 157, 175. 193, 195, aoi, 313

Enunciation (r/Mfro^it), one of formal di-
▼isions of a proposition 139-30

Epicureans, objection to i. ao 41, 387 :
Savile on, 387

Equality, in sense different from that of
congruence ( = '* equivalent," Legendre)
337-8: two senses of equal (i) "divisibly-
equal" (Hilbert) or '* equivalent by sum"
(Amaldi), (3) ** equal in content" (Hilbert)
or '* equivalent bv difference" (Amaldi)
338 : modem dennition of, 338

Eratosthenes i : contemporary with Archi-
medes I, 3 : 163

Errard, Jean, de Bar-le-Duc 108

Erydnus 37, 390, 339

Euclid : account of, in Proclus' summary i ;
date 1-3 : allusions to in Archimedes i :
(according to Produs) a Platonist 3 : taught
at Alexandria 3 : Pappus on personality
of, 3: story of (in Stobaeus) 3 : not "of
Megara** 3, 4: supposed to have been
bom at Gela 4 : Arabian traditions about,
4,5! "of Tyre" 4-6: "of m" 4. 5«-:
Arabian derivation of name ("key of
geometnr") 6 : Steminis^ ultimate aim of,
3, 1 1 5-0: other works. Conies 16, Psm-
daria 7, Data 8, M3, 141, 385, 391, On
divisions (of figures) 8, 9, Porisms fO-15,
Snrfaet-loci 15, 1*6, Pkamonuna 16, 17,
Optics 17, Elements pf Music or SecHo
Canonis 17: on "three- and four-line
locus" 3 : Arabian list of works 17, 18 :
bibliography 91-113

Eudemus 39: On the A^gle ^, 38, 177-8:
I/istory of Geometry 34, 35-8, 378, 195,
304. 3»7. 3«o, 387

Eudoxus I, 37, 1 10: discoverer of theory
of proportion as expounded generally in
Bks. v., VI. 137, 351 : on the golden
section 137 : fowider of method of ex-
haustion 334 : inventor of a certain curve,
the kipMbede^ horse-fetter 163: possibly
wrote ipkaerica 17

Enler, Lmnhard 401

Etttodtts 35, 35, 39, 143, 161, 164, 359, 317,

3«9> 330, 373
Exterior and interior (of angles) 363, 380
Extremity^ w^% 183, 183

Falk, H. 113

al-FaiadI 8«i., 90

Figure^ 9% viewed by Plato 183, by Aristotle
183-3, by Eudki 183: aooording to Posi-
donitts is confining houncUuy ooXy 41, 183:
figures bounded by two lines classified 187 :
emgle-less (dtyiirMr) figure 187

Fisures, printing of, 97

FArist 4M., 5«i., 17, 31, 34, 35, 37: list of
Euclid's works in 17, 18

Ftnaeos, Orootius (Oronce Fine) loi, 104

H. B.

FUuti, Vincenzo 107

Florence MS. Laurent, xxviii. 3, (F) 47

Flussates, see Candalla

Forcadel, Pierre 108

Fourier 173-4

Frankland, W. B. 173, 199

Frischauf 174

Gartz 911.

Gauss 173, 193, 194, 303, 319. 331

Geminus: name not Latin 38-9 : title of work
(0<XoiraX(a) quoted from by Produs 39:
elements of astronomy 38: comm. <m Posi-
donius 39 : Produs* obligations to, 39-43 :
on postulates and axioms 133-3: on theo-
rems and problems 138: two classifications
of lines (or curves) 160-3: on homoeo-
meric {uniform) lines 163: on '* mixed"
lines (curves) and suriiMres 163 : classifica-
tion of surfaces 170^ of angles 178-9:
on parallels 191: on Postulate 4, 300:
on stages of proof of theorem of i. 33, 317-
30: 31, 37-8, 37, 44, 45, 133 If., 303,
365, 330

Geometrical algebra 373-4 : Eudid's method
in Book 11. evidently the dassical method
37 3 : preferable to semi-algebraical method

377-8

Gherard of Cremona, translator of Elements
93-4: of an-Naii1zI's commentary 33, 94 :
of tract De divisionibus 9

Giordano, Vitale 106, 176

Given, d€9otUrot, different senses, 133-3

Gnomon : literallv ** that enabling (something)
to be Anown" 64, 370 : successive senses of,
(i) upright /9fan6^ of sundial 181, 185, 371-
3, introduced into Greece by Anaximander
370, (3) carpenter's square for drawing
right angles 371, (3) figure placed round
square to make larger square 351, 371,
Indian use of gnomon in this sense 363,
(4) use extendedby Eudid to paralldocrams
37 > > (5) ^ Heron and Theon to any Sgarts
371-3: Euclid's method of denoting in
figure 383 : arithmetical use of, 358-60, 371

*' Gnomon- wise" (card yvti/Mfa), old name
for perpendicular (ciC^crot) 36, 181, 373

Gorland, A. 333, 334

"Golden section "s section in extreme and
mean ratio 137: connexion with theory of
irrationals 137

••Goose's foot'^ {pes anseris), name
Eud. III. 7, 99

Gow, James 135 n.

Gracilis, Stephuius 101-3

Grandi, Guido 107

Gregory, Davkl 103-3

Gregory of St Vincent 401, 404

Gromatici 91 »., 95

Grynaeus loo-i

al-Haitham 88, 89

al-Hajjftj b. YCisuf b. Matar, translator of the

Elements 33, 75, 76, 79, 80, 83, 84
Halifax, William 108, no

a;

for

4i8

ENGLISH INDEX

Hmlliwell 05 m.

Hankel, H. 139, 141. a3«, 134, 344, 354

JfarmcMsca of Ptolemy, Comm. on, 17

harmony, Iniroduetum to, not by Euclid 17

HirQn ar-RashId 75

al-Hasan b. *UbaidalULh b. Solaim&n b.

Wahb 87
Haaff, J. K. F. 108
" Heavy and Light,** tract on, 18

Heiberg, J. L. passim
Helix, cyfin • * • '

i6a, 319, 330

/Ondrical 16 1

Helmhoftz 216, 327

Henrici and Trentlein 313, 404

Henrion, Denis 108

H^ri^one, Pierre 108

Herlin, Christian 100

Hermotimos of Colophon i

Herodotus 37 »., 370

•*Heromides** 158

Heron of Alexandria, mt^AankuSt date of
20-1 : Heron and Vitruvius lo-i : com-
mentliry on Euclid*s Elements 30-4 :
direct proof of l. 25, 301 : comparison of
areas of triangles in i. 14, 334-5 : addi-
tion to 1. 47, 366-8 : apparently originated
semi-algebraical methoa of proving theo-
rems of Book II. 373, 378 : 137 «., 159,
163, 168, 170, 171-2, 170, 183, 184, 185,
188, 189, «M. M3, «43, «53, 185, 187,

«99. SSh 360, 371 > 405f 407. 408
Heron, Proclus instructor 19
''Hemndes" 156
Hieronymus of Rhodes 305
Hilbert 157, 193, 301, 438-31, 349>3t3t 3^^
Hipparchus 411., 30 «.
Hippias of Elis 49, 365-6
Hippocrates of Chios Sn., 39, 35, 38, 116,

135. 1)6 n,, 386-7
Hifp&Ude (trrov W3if), a certain curve used

by Eudoxus 163-3, 176
Hoffmann, Heinrich 107
Hoffinann, John Jos. len. 108, 365
Holtxmann, Wilhelm (Xylander) 107
Hotnceomific (uniform) lines 40, 161, 163
Hoppe, £.31
Hornlike (angle), ictparoei^ 177, 178, 183,

96s
Horsley, Samuel 106
Hottel, J. 319
Hudson, John 103
Hultsch, F. 30, 339, 400
Ijunain b. Ish2q al'ibidi 75
Hypotheses, in Plato 133 : in Aristotle 118-

30: confused by Proclus with de6nitions

13 1-3 : geometer's hjrpotbeses not false

(Aristotle) 119
HjTpothetical construction 199
Hypsides 5 \ author of Book xiv. 5, 6

lamblichus 63, 83

Ibn al.*AmId 86

Ibn al-Haitham 88, 89

Ibn al-LobCidI 90 •

Ibn Rihawaibi al-AijinI 86

Ibn Sinft (Avioenna) 77, 89

«• Iflaton '* 88

Incomposite (of lines) 160-1, (of surfiices)f 70

Indivisible lines (dro/ioc ypofipMl), theory of,
rebutted 368

Infinite, Aristotle <m the, 333-4: infinite
division not assumed, but proved, by geo-
meters 2^

Infinity, parallels meeting at, 193-3

Ingrami, G. 175, 193, 195, 301, 337-8

Interior and exterior (of angles) 363, 380 :
interior and opposite angle 380

Interpolations in tne Elements before Theon's
time 58-63 : by Theon 46, 55-6 : I. 40
interpolated 338

Irrational I discovered with reference to J 2
351 : claim of India to priority of dis-
covery 363-4 : ** irrational diameter of 5 "
(Pythagoreans and Plato) 399-400 : ap-
proximation to iji by means of^ "side-*'
and ** diagonal-** numberB^99-40i : Indian
approximation to J% ^fit 363-4: smi-
oraered irrationals (ApoUonius) 43, 115:
irrational ratio {dpfnfret Xiyos) 137

Isaacus Monachus (or Aigyrusl 73-4, 407

Ishfta b. Hunain b. Ishftq al-Ibidl, Abu
Vaqikb, translaticm of Elements by, 75-80,
83-4

Ismail b. Bulbul 88

Isoperimetric (or isometric) figures : Pappus
and Zenodorus on, 36, 37, 333

Isosceles {fffo9K€kify 187: of numbers (= even)
188 : isosceles right-angled triangle 353

Jakob b. Machir 76
al-Jauharl, al-*Abbfts b. Sa*Id 85
al-Jayyftnl 90

{oannes Pediasimus 73-3
unge, G., on attribution of theorem of i. 47
and discovery of irrationals to Pythagoras
351

KiUtner, A. G. 78, 97, loi

al-KaritblSI 85

Kitvftyana Sulba-Sutra 360

Keiu, Jphn 105, iio-ii

Kepler 193

al-KhUin, Abu Ja*far 77, 85

Killing, W. 194, 319, 335-6, 335, 343, 373

al-Kindl 5 ii., 86

Klamroth, M. 75-84

Klttgel, G. S. 313

Knesa, Jakob 113

Knoche asii., 33*., 73

KroU, W. 399-400

al-K&hl 88

Lambert, J. H. 313-3
Lardner, Dionysius 113, 346, 350, 398, 404
Lascaris, Constantinus 3
Leading theorems (as distinct from eMfsvrr^
357 : leading variety of conversion 356-7
Leeke, John no
Lef^vre, Jacques 100

Legendre, Adrien Marie 119, i(^, 913-9
Lobnis 145, 169, 176* 194

H

ENGLISH INDEX

419

i

Leiden MS. 399, i of al-Hajjij and an-
NairizI 33

Lemma 114: meaning 135-4 : lemmas inter-
polated 59-60, especially from Pappus 67

LcKxlamas of Thasos 36, 134

Leon 116

Linderup, H. C 11^

Line: Platonic definition 158: objection of
Aristotle 158: '* magnitude extended one
way " (Aristotle, . ** Heromides"} 158 :
" divisible or continuous one way* (Aris-
totle) 158-9: <* flux of point " 159: Apol-
lonius on, 159 : classification of lines, Plato
and Aristotle 159^-60, Heron 159-60,
Geminus, first classification 160-1, second
161 : straight (d^^cca), curved (KafiwAXii),
circular (re/H^c/M^t), spiral-shaped {iXxKo-
ci^), bent (irciraAi/i^'i;),' broken (ireirXa-
vfUvri), round {ffTpcyy6\ot) 159, composite
{<rC-p$€Tot), incomposite (dffMerot), "form-
ing a flgure " (tf'x^MaTovoiouo'a), determinate
{Cipia/jJpyi)^ indeterminate {(Upirros) 160:
"asymptotic" or non-secant {iffi^/iwriarot),
secant {avfirrtarSs) x6i : simple, "mixed"
1 61-3 : homoeom^rie (uniform) 161-3:
Proclus on lines without extremities 165 :
loci on lines 339, 330

Limar, loci 330: problems 330

Lionardo da Vinci, proof of I. 47 365-6

Lippert 88 n,

Lobachewsky, N. I. i74-5> 313, 319

Locus-theorems (roviird Bttapf^yMJO^ and loci
{^ToC^ : locus defined by Proclus 339 :
loci likened by Chrysippus to Platonic
ideas 330-x : locus-theorems and lod (i) on
Unes (a) plcute loci (straight lines and
circles) (b) solid loci (conies), (3) on sur-
faces 339 : corresponding distinction be-
tween plane and solid problems, to which
Pappus adds Hnear proolems 330 : further
distinction in Pappus between (i) i^xriKol
(3) 9i€^8ucol (3) dpaoTpo^Kcl t6ww, 330:
Proclus regards locus in I. 35, III. 3i, 31
as an area which is locus of area (pai:allelo-
gxam or triangle) 3^0 .

Lc^cal conversion, distinct from geometrical
156

Logical deductions 356, 384-5, 300 : logical
equivalents 309, 314-5

Lorenz, J. F. 107-8

Loria, d-ino 7 if., 10 if., 11 if*, 13 if.

Luca Paciuolo 98-i), 100

Lundgren, *F. A. A. 113

Machir, Jakob b. 76

Magni, Domenico 106

Magnitude: common definition vicious 148

al-M&hftnl 85

al-Ma'miin. Caliph 75

Mansion, P! 319

al-Manfiir, Caliph 75

Manuscripts of Elements 46-51

Blartianus Capella 91, 155

Martin, T. H. 30, 39 m., 30 n,

Mas*&d b. al-Qass al-Ba^didl 90

Maximus Planudes, scholia and lectures on
Elements 73

m^^fMirssaxis 93

Mehler, F. G. 404

Meier, Rudolf 31 m.

Menaechmus: story of M. and Alexander i :
on elements 114: 117, 135, 133 if.

Menelaus 3i, 33: direct proof of I. 35 ^00

Middle term, or cause, m geometry, illus-
trated by III. 31 149

MiU, J. S. 144

*' Mixed*' (lines) 161-3: (surfaces) 163, 170:
different meanings of " mixed '* 163

Mocenigo, Prince 97-8

MoUw^de, C. B. 108

Mondor^ (Montaureus), Pierre 103

Moses b. Tibbon 76

Motion, in mathematics 336 : motion tenth-
out deformation considered by Helmholtz
necessary to geometry 336-7, but shown
by Veronese to ht petitio principii 336-7 •

Mufler, J. H. T. 189

MUller, J. W. 365

Muhammad (b. ^Abdalbftql) al-BagdidI,
translator of De divisionibus 8 if ., 90,
no

Muh. b. Ahmad Abti 'r-Raihin al-Blrtlnl 90

Muh. b. Ashraf Shamsaddin as-Samarqandl

M^. b. is& Abu *AbdalULh al-Mahini 85
Munich MS. of enunciations (R) 94-5
al-Mutawakkil, Caliph 75
Musi b. Muh. b. MahmQd Qidlzide ar-

Rdml 5 If., 90
MusiCt Elements of {Sectio Canonis), by

Euclid 17
al-Musta'$im, Caliph 90

an-NairlzI, Abu '1 *Abbfts al-Fadl b. Qfttim,
31-34, 85. 184. 190, 191, 195, 333. 333,
358, 370, 385, 399, 303, 336, 364, 367,

^369. 373. 405. 408

Napoleon 103

Nafiraddin at-Tusi 4, 5*., 77, 84, 89,
308-10

Naflf b. Yumn (Yaman) al-Qass 76, 77, 87

Neide, J. G. C. 103

Nicomachus 93

Nicomedes 43, 160-1, 365-6

Nipsus, Marcus Tunius 305

Nominal and r«s/ definitions : see Definitions

Obfection (Iroro^if), technical term, in
geometry 135, 357, 360, 365 : in Ipgic
(Aristotle) 135

Oblong 151, 188

Oenopides of Chios 34, 36, 136, 371, 395,

371
Ofterdinger, L. F. 9 if., 10
Olympiodorus 39
Oppermann 151
Opiics of Euclid 17
Oresme, N. 97 *
Orontios Finaens (Oxonce Fine) 10 1, 104
Ozanam, Jaques 107, 108

430

ENGLISH INDEX

Paduolo, Lnca 9^t loo

Pamphile 317, 319

Pappus : contrasts Euclid and ApoUonius 3 :
on Euclid's Ptrisms 10-14, Surfda-lici
15, 16, Data 8: on Treasury of Analysis
8, 10, II, 138: commentaiy on EUments
14-9, partly pretenred in scholia ^',
endence of sdiolia as to Pappus' text
66-7: lemmas in Book x. interpolated
from, 67 : on Analysis and Synthesis 138-9,
141-s: additional axioms by, 95, 193, 994,
939 : on converse of Post. 4 95, 90i :
proof of I. 5 by, 954 : extension of i. 47
366: semi-algebraical methods in 373,
378 : on loci 399, 330: on conchoids 161,
966 : on auadratrix 966 : on isoperimetric
fibres 90, 97, 333: on paradoxes of
fiycinus 97, 990: 17, 39, 133 «., i37.
I5i> <«5» 388, 39i> 401

Paj^mis, Herculanensis No. 1061 50, 184:
Oxrrhynchus 50: Fayum 51, 337, 338:
Rhind 304

Paradoxes, in geometry 188: of Erydnus
97, 99o» 399: an ancient "Budget of
Ptoidoxes*^ 399

Purallelogram (s parallelogrammic area),
first introduced 395 : rectangular parallelo-
gram 370

Parallels: Aristotle on, 190, 191-9: defini-
tions, by '^Aganis** 191, 1)y Geminus 191,
Poskiomus 190, Simplicius 190: as equi-
distants 190-1, 194: direction-theory of,
10 1-9, 104 : definitions classified 199-4 :
Veronese's definition and postulate 194 :
Pandlel Postulate, see Postulate 5 :
Legendre*s attempt to establish theory of
913-9

Paris MSS. of Elements, (p) 49: (q) 50

Pasch, M. 157, 998, 950

"Peacock's tail,** name for iii. 8 99

PMliasimus, Joannes 79-3

Peithon 903

Pdetarius (Jacques Peletier) 103, 104, 949,

407
Pena 104
Perpendicular (cd^crot) : definition 181 :

"plane" and "solid ^ 979: perpendicular

and obliques 991
PerMus 49, 169-3
Pesch, J. G. Tan, De Proclifentihus 93 sqq.,

t9«.
Petrus Montaureus (Pierre Mondor^) 109
Peyrard and Vatican MS. 190 (P) 46, 47,

•103: 108
Pfldderer, C. F. 168, 998
Phaenetnena of Euclid 16, 17
Philippus of Mende i, 116
Phillips, George ii9
Philo of Byzantium 90, 93 : proof of X. 8

963-4
^iloUus 34, 351, 371, 399
Philoponus 45, 191-9
Pirckenstein, A. £. Burkh. von 107
Plane (or plane sarfiice) : Plato's definition

of, 171: Produs' and Simplidus* inter-

pretation of Euclid's def. 171 : possible
origin of Euclid's def. 171 : Archimedes*
assumption 171, 179: other andent defini-
tions of, in Plrodus, Heron, Theon of
Smjrna, an-NaiiisI 171-9: "Simaon's"
definition and Gauss on 179-3: Crdle^s
tract on, 179-4: other definitions by
Fourier 173, Deahna 174, J. K. Becker
174, Leibniz 176, Beez 176: evolution of,
by Bolyai and Lobadiewsky 174-5:
Enrioues and Amaldi, Ingrami, Veronese
and Hilbert on, 175

" Plane lod " 399-30: PUme Loci of Apol-
lonius 14, 959, 330

"Plane problems ** 399

Planndes, Maximus 79

PUto: I, 9, 3, 137, 155-6, 159, 184, 187,
903, 991 : supposed inventicm of Analysis
by, 134: def. of straight line 165-6 : def.
of plane surface 171 : generation of cosmic
figures by putting to^^er triangles 996 :
rule for rational ng^t-angled triangles 356,
3$7* J59> 3<^ 3^5* "niHoiud diameter
of 5 399

"Platonic'* inures 9

PlaTfair, John 103, iii: "Playfidr's"
Axiom 990: used to prove i. 99, 319, and
Eud. Post. 5, 313: comparison dr Axiom
with Post. 5, 313-4

P^y «o, 333

Plutarch 91, 99, 37, 177, 343, 351

Point: Pythaj»rean definition of, 155: inter-
pretation otjEudid's definition 155: Plato's
▼iew of, and Aristotle's criticism 155-6:
attributes of, according to Aristotle 156:
terms for ((myiufi, 9yii»Mtm) 156: other
definitions by '* Herundes," Posidonius
156, Simplicius 157 : negpttiTe character of
Eudid's def. 156: is it sufficient? 156:
motion of, produces line 157: an-NairlzI
on, 157 : modem explanations by abstrac-
tion 157

Polybius 331

Polygon : sum of interior angles (Produs'
I^ooQ 399 : sum of exterior an«es 399

Pmism : two senses 13 : (i) = corollanr 134,
978-9: interpolatea Porisms (corollaries)
60-1, 381 : (9) as used in /^imiMf of Eudid,
distinguished from theorems and problems

10, II : account of the Porisms given \yy
Pappus 10-13: modem restorations 1^
Simson and CHiasles 14 : views of Hdberg

11, 14, and Zeuthen 15

Porphjrry 17: commentary on Eudid 94:
Symmikta 94. 34, 44 : 136, 977, 983, 987

Posidonius, the Stoic so, 91, 97, 98 is., 189,
107 : book directed against the Epicurean
2^0 34, 43 : on parallds 40, 190: defini-
tion of Jignre 41, 183

Postulate, distingutshed from axiom, by
Aristotle 118-^, by Produs (Geminus
and "others") 191-3: firom hypothesis,
by Aristotle 190-1, by Produs iti-«:
postulates in Ardiimedes iso^ 1*3:
Euclid's view of, leoondleaUe wiUi

ENGLISH INDEX

431

i

V

Aristotle's 119--10, 114: postulate do not
coofiDe tts to rule and compass 1 34 : Postu-
lates I, a, significance of, 195-6: famous
** Postulate of Archimedes" 934

Postulate 4 : significance of, 100 : proofs of,
resting on other postulates loo-i, 131:
converse true only when angles rectilineal
(Pappus) 101

Postulate 5: due to Euclid himself 401:
Proclus on, 303-3: attempts to prove,
Ptolemy 304-6, Proclus 306-8, Na$lraddln
at-TiisI 30&-I0, Wallis 310-1, Saccheri
3 1 1-3, Lambert 913-3: substitutes for,
" Plavfair's" axiom (in Proclus) 330, others
by Proclus 307, 330, Posidonius and
Geminus 330, Legendre 313, 314, 330,
Wallis 330, Camot, Laplace, Lorenz,
W. Bolyai, Gauss, Worpitzky, Clairaut,
Veronese, Ingram! 330 : Post. 5 proved
from, and compared with, " Playfair*s **
axiom 313-4: I. 30 is logical equivalent

of, 330

Potts, Robert 113, 346

Prime (of numbers) : two senses of, 146

Principles, First 117-134

Problem, distinguished from theorem 134-8:
problems classified according to number of
solutions (a) one solution, ordered (rtTay-
lUwa) {b) a definite number, intermediate
{liha) (c) an infinite number of solutions,
unordered (dtnurra) 138 : in widest sense
anything propounded (possible or not) but
generalfy a construction which is pouible
128-9: another classification (i) problem
in excess (rXcord^pr), asking too much 139,
(3) deficient problem (AXiWt rp^/SXif/ui),

. giving too little 139

Proclus: details of career 39-30: remarks
on earlier commentators 19, 33, 45 : com-
mentary on Eucl. I, sources of, 39-45,
object and character of, 31-3 : com-
• mentary probably not continued, thoufh
continuation intended 33-3 : books
quoted by name in, 34: famous "sum-
nuiy** 37-8: list of writers ouoted 44:
his own contributions 44-5: cnaracter of
MS. used bv, 63, 63: on the nature of
elements and things elementary 11 4-6: on
advantage of Euclid's Elements^ and
their object 11 5-6 : on first principles,
hypotheses, postulates, axioms 13 1-4: 00
difficulties in three distinctions between
postulates and axioms 133: on theorems
and problems 134-9: attempt to prove
Postulate 5 306-8 : commentaiy on Plato's
Refubbc^ allusion in to '*side-" and
"aiagooal-" numbers in connexion with
Eucl. II. 9, 10 399-400

Proef (dvMi^t), necosaiy part of pro-
position 130-30

ProposiHon^ formal divisions of, 149-13 1

Protarchus 5

FkIIus, Michael, scholia by, 70, 71

Pseudaria of Euclid 7: Pseudograpkemaia 7 n,

Pseudoboethius 93

Ptolemy I.: i, 3: story of Euclid and
Ptolemy t

Ptolemy, Claudius 30 n, : Harmonica of, and
commentary on 17: on Parallel- Postulate
9^ ^'t 34t 43t 45 '' attempt to prove it 304-6

Pythagoras 4 n., 36 : supposed discoverer of
the irrational 351, of application of areas
343-4, of theorem of l. 47 343-4. 350-4;
storv of sacrifice 37, 343, 350: probable
method of discovery of i. 47 and proof of,
353-5: suggestions by Bretschneider and
Hankel 354, by Zeuthen 355-6: rule for
forming right-angled triangles in rational
numbers 351, 356-9, 385

Pythagoreans 19, 36, 155, 188, 379: term
for surface (xp^) 169 : angles of triangle
equal to two right angles, theorem and
proof 317-30: mree polygons which in
contact fill space round point 318 : method
of application of areas (mduding exceeding
and falling-short) 343, 384, 403 : gnomon
P3rthagorean 351: ** rational'* and "ir-
rational diameter of 5 " 399-400

Q&dlz&de ar-RumI 5^., 90

Q.E.D. (or F.) 57

al-QiftI 4 If., 94

Quadratic eouation, geometrical solution of,
3^3-5t 386-8 : solution assumed by Hippo-
crates 386-7

Quadratrix 365-6, 330

Quadrature (rrrpaywtoiiM), definitions of, 1 49

Quadrilaterals, varieties o^ 188-90

Quintilian 333

Qustft b. Luqi al-Ba*labakkI, translator of
••Books Xiv, XV '* 76, 87, 88

Radius, no Greek word for, 199

Ramus, Petrus (Pierre de la Run^) 104

Ratdolt, Erhard 78, 97

Rationed (^6t) : (of ratios) 137 : ** rational
diameter of ^ *' 399-400 : rational right-
angled triangles, see ri^ht-angled trian^es

Rau<mfuss, see Dasypodius

Rausenberger, O. 157, 175, 313

ar-RftzI, Abu Yusuf Yaqfib b. Muh. 86

Rectangle: = rectangular paraUelognun
370 : •• rectangle contained by " 370

Rectilineal angle : definitions classified 179-
81: rectilineal figure 187: "rectilineal
segment" 196

Peductio ad absurdum 134: described by
Aristotle and Proclus 136 : synonyms for,
in Aristotle 136: a variety of Anal^is
140: by exhaustion 385, 393: nommal
avoidance of 369

Reduction (draTc^Ti^), technical term, ex-
plained by Aristotle and Proclus 1^5:
first *' reduction " ofa difficult construction
due to Hippocrates 135

Regiomontanus (Johannes Miiller of Konigs-

^ *>««) 93* 9<^» »oo
Reyher, Samuel 107
Rhaeticus loi
Rhomboid 189

433

ENGLISH INDEX

Rhombus, meaning and derivation 189
Riccardi, P. 96, 11 3, loi
Riemann, B. a 10, 373, 374, 480
Ri^ht angle : definition 18 x : drawing straight
hne at ri^t angles to another, Apollonius'
construction for, 370: construction when
drawn at extremity of second line (Heron)
170 •
Rieht-angled triangles, rational: rule for
finding, by Pythagoras 356-9, by Plato
35^» 357» 359« S^Of 385 : discovery of rules
by means of gnomons 358-60 : connexion
of rules with Eud. 11. 4, 8, 360 : rational
rieht-angled triangles in Apastamba 361,

Roth 357-8

Roucbe and de Comberousse 313
Rudd, Capt. Thos. no
Ruellius, Joan. (lean Ruel) 100
Russell, Bertrand 327, 349

Saccheri, Gerolamo 106, 144-51 '67-8,

185-6, 194, 197-8, 300-1

Said b. Masiid b. al-Qass 90

Sathapatha-Brfthmana 363

Savile, Henry 105,* 166, 345, 350, 363

StaUui (rcaXip6t or SKoXtirtfi) 187-8: of
numbers (= odd) 188: of cone (ApoUonius)
188

Schessler, Chr. 107

Scheubel, Joan, loi, 107

Schiaparelli, G. V. 163

Schlttssd, Christoph, ste Clavius

Schmidt, Max C. P. 304, 319

Schmidt, W., editor of Heron, on Heron's
date 30-I

Scholia to Elements and ifss. of 64-74:
historical information in, 64 : evidence in,
as to text 64-5, 66-7 : sometimes inter-
polated in text 67: classes of, "Schol.
Vat." 65-^ "SchoL Vind." 6^70: misceU
laneous 71-4 : "Schol. Vat" partly derived
from Pappus* commentary 66: many
scholia i)amy extracted from Produs on
Bk. I. 66, 6j^, 73 : numerical illustrations
in, in Greek and Arabic numerak 71:
scholia by Psellus 70-1, by Mazimus
Planudes 73, Joannes Pediasimus 73-3:
scholia in Latin published by G. VaUa,
Commandinus, Conrad Dasypodius 73 :
scholia on Eud. 11. 13 407

Schooten, Franz van 108

Schopenhauer 337, 354

Schotten, H. 167, 174, 179. i9«-3t 903

Schumacher 331

Schur, F. 338

Schwdkart, F. K. 3x9

Sdpio Vegius 99

SecHo Camomis by Eudid 17

Section (ro/n():s/0f#f/ «^ section x7o, 171,
383: "M^scction**=*'^goldensection'V.».

Segment of drcle, angle of^ 35^ : segment
less than semicircle called ^t 187

Semidrde 186: centre of, 180: angle ef^
183, 353

Seqi ^^

Serenus of Antinoeia 303

Serle, George no

Setting-cut {Mwkt)^ one of formal divisions
of a proposition 139 : may be omitted 130

Sextus Empiricus 63, 63, 184

Shamsaddin as-Samarqandl 5ii., 80

"Side-** and "diagonal-" numbers, described
398-400: due to Pythagoreans 400: con-
nexion with Eucl. II. 9, 10 398-400: use
for approximation to »sji 399

Sigboto 04

"Similar^' (= equal) angles 183, 353: "simi-
lar** numbers 357

Simon, Max 108, 155, 157-8, 167, 303,

Simplicius: commentary on Euclid 37-8:
on lunes of Hippocrates 39, 35, 386-7:
on Eudemus' style 35, 38: on parallels
190-1: 33, 167. 171, 184, 185, i97f «03,
^93, 334

Simson, Robert: on Eudid*s Porisms 14:
on "vitiations** in Eletnents due to Theon
46, 103, 104, fo6, 1x1, 148: definition
of plane 173-3: 185, 186, 355, 359, 387,

«93. »^. 3«a. 3«8. 384* 387. 403

Sind b. 'AH Abii VT«iyib 86

Smith and Bryant 404

"Solid loci** 3«9» 33o: Solid Loci of Aris-
taeus 16, 339

"Solid problems'* 339, 330

Speusippus 135

SphaerUa^ early treatise on, 17

Spiral, "single-turn** 133-311., 164-5: in
Pappus = cylindrical helix 165

Spiral of Archimedes 36, 367

Spirt (tore) or Spiric surface 163, 170^
varieties of 163

Spiric curves or sections, discovered by
Perseus 161, 163-4

Steenstra, F^bo 109

Steiner, Jakob 193

Steinmets, Moritz loi

Steinschndder, M. 811., 76 sqq.

Stephanus Gracilis 101-3

Stephen Clericus 47

Stobaeus ^

Stoic "axioms" 41, 331 : ilhistrations (fcf)r*
IMxa) 339 .

Stolz, O. 338

Stone, E. 105

Straight line: pre-Eudidean (Platonic) de-
finition 165-6: Ardiimedes* astumfHon
respecting, 166 : Euclid*s definition, mter-
preted by Proclus and Simplidus 166-7 :
language and constructicm of, 167, and
conjecture as to origin 168: other defi-
nitions 168-9, ^ Heron 168, by Leib-
niz 169, by Legendre 169: two straight
lines cannot enclose a space 195-6, can-
not have a common segment 196-9: one
or two cannot make a figure 169, 183:
division of straight line into any number
of equal puts (an-NaixIzI) 336

Stromer, Marten 113

c

J

ENGLISH INDEX

4*3

\

Studemund, W. paw.

St Vincent, Gregory of, 401, 404

Subtend^ meaning ind construction 949,

«83, 350

Suidas 370

Sulaim&n b. 'U^ma (or Ocjba) 85, 90

Superposition: Euclid's dislike of method
of, 135, 949: apparently assumed b^ Aris-
totle as legitimate 116 : used by Arclumedes
415 : objected to by Peletarius 349 : no use
theoretically, but merely i^xmi^tA practical
test of equality 337 : Bertrand Russell on,
337, 349

Surface: Pythagorean term for, xji^k (= col-
our, or skin) 169: terms for, in Plato and
Aristotle 169: ixt^pua in Euclid (not
MTfdov) 169: alternative deBnition of, in
Aristotle 170: produced by motion of
line 170: divisions or sections of solids
are surfaces 170, 171: classifications of
surfaces by Heron and Geminus 170: com-
posite, incomposite, simple, mixed 170:
spiric surfaces 163, 170: homoeonuric
(unirorm) 170: spheroids 170: plane sur-
face, see plane : loci on suihces 339, 330

Sutf ace-loci of Euclid 15, 16, 330: Pappus'
lemmas on, 15, 16

Suter, H. 811., 9«., i7«., 18 if., 35 j»., 78 j».,

Suvoroff, Pr. 113
Swinden, J. H. van 169
Synthesis, see Analysis and Synthesis
Syrianus 30, 44, 176, 178

Tacquet, Andr^ 103, 105, 11 1

Tftittiriya-Sa]phit& 363

Tannery, P. 711., 37-40, 44, 160, 163, 331,

333, 334, «35» «3«» 305t 353
Ta'rikh al Hukamd 411.
Tartaglia, Niccol6 3, 103, 106
Taurinus, F. A. 319
Taurus 63, 184

Taylor, H. M. 348, 377-8. 404
Taylor, Th. 359
Thabit b. Qurra, translator of Elements

9«.. 4«. 75-8o» 83, 84f 87, 94: proof of u

47 364-5
Thales 36, 37, 185, 353, 353, 378, 317,

318, 319: on distance of ship hom shore

^304-5

Theaetetus i, 37

Theodorus AntiochiU 71

Theodoras Cabasilas 73

Theodorus MetochiU 3

Theognis 371

Theon of Alexandria: edition of Elements
46: changes made by, 46: Simson on
"vitiations'* by, 46: principles for detect-
ing his alterations, by com[)arison of P,
ancient papyri and "Theonine" ifss. 51-
3: character of changes by, 54-8

Theon of Smyrna 173, 357, 358, 371, 308

Theorem and problem, distinguished By
Speusippus 135, Amphinomus 135, 138,
Menaccnmns 135, ZenodotiiSy Ptondooiiis

136, Euclid 136, Carpus i37» 1^8:
views of Proclus 137-8, and of Geminus
138: "general" and ** not-general" (or
partial) theorems (Proclus) 335

Theudius of Magnesia X17

Thibaut, B. F. 331

Thibaut, C: On 6ulvasutras 360, 363-4

Thompson, Thomas Perronet 113

Thucydides 333

Tibbon, Moses b. 76

Tiraboschi 9411.

Tittel. K. 39

Todhunter, I. 11 3, 189, 346, 358, 377,

983, a93« «98. 307

Tonstall, Cuthbert 100

Tore 163

Transformation of areas 346-7, 410

Trapezium : Euclid's definition his own 189 :
further division into trapezia and trape-
zoids (Posidonius, Heron) 18^-90 : a
theorem on area of parallel-trapezium

338-9
Treasury of Analysis ((UaXu6ficyot rArot)

8, 10, II, 138

Trendelenburg 146M., 148, 149

Treutlein, P. 358-60

Triangle: seven species of, x88 : "four-
sided" triangle, called also "barb-like"
(d«idoctd^f) and (by Zenodorus) coiXo^a-
9W9 37, 188: constraction of isosceles and
sodene triangles 343

Trisection of an angle 365-7

at-fusl, see Naflraddin'

Unger, E. S. 108, 169

Vachtchenko-Zakhartchenko 1x3

Vailati, G. 144 m., 145 «>•

Valerius Maximus 3

Valla, G., De expetendis et fugiendis rebus

73> 98
Van Swinden 169
Vatican MS. iqo (P) 46, 47
Vaux, Cam de 30
Verona palimpsest 91
Veronese, G. 157. 168, 175. >8o, 193-4,

195. «oi, 336-7, ««8, 349, 338
Vertical (angles) 378
Viennese MS. (V) 48, 49
Vind, Lionardo da 365-6
Vitravius 353: Vitravius and Heron 30, 31
Viviani, Vincenzo 107, 401
Vogt, Heinrich 360, 364
Vooght, C. J. 108

Wachsmuth, C 3311., 73
Wallis, John 103: edited Comm. on Ptol-
emy's Harmonica 17: attempt to prove

Post 5 31^1

Weber (H.) and Wellstein (J.) 157
Weissenbom, H. 78«i., 9311., 94 m., 95,

96, 97 «.
Whiston, W. Ill
Williamson, James iii, 393
Witt, H. A. 115

4^4

ENGLISH INDEX

Woqpcke, F., disooTered De dhisiamibus in
Arabic and published translation 9: on
Pappus* oofmnentary on EUmtnis 35, 66,
77: 85«., 86. 87

Xenocrates 968
Ximenes, Leonardo 107
Xjlander 107

Yahyft b. Kh&lid b. Barmak 75
Yahyi b. Muh. b. *Abdin b. Abdalwihid
(ibn al-Lubudl) 90

Yrinus= Heron 13

YCihanni b. Yfisuf b. al-Ij[irith b. d-Bitrlq
al-Qass 76, 87

Zamberti, Bartolomeo 98-100, loi, 104,

106
Zeno the Epicurean 34, 196, 197, 199,

141
Zenodorus 16, 97, 188, 333
Zenodotus 196
Zeuthen, H. G. 15, 139, 141, 14611., 151,

355-6, 360, 363, 387, 398, 399

• V

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